Theorem 1 (Burns-Masur-Wilkinson)Suppose that:Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously. Denote by the metric completion of and the boundary of .

- (I) the universal cover of is
geodesically convex, i.e., for every , there exists an unique geodesic segmentinconnecting and .- (II) the metric completion of is
compact.- (III) the boundary is
volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfiesfor every .

- (IV) has
polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial boundfor every .

- (V) has
polynomially controlled injectivity radius, i.e., there are constants and such thatfor every (where denotes the injectivity radius at ).

- (VI) The
first derivative of the geodesic flowispolynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :Then, the Liouville (volume) measure of is finite, the geodesic flow on the unit cotangent bundle of is defined at -almost every point for all time , and the geodesic flow is

non-uniformly hyperbolic(in the sense of Pesin’s theory) andergodic.

Actually, the geodesic flow is Bernoulli and, furthermore, its metric entropy is positive, finite and is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of counted with multiplicities).

More precisely, we proved in the previous post of this series that a geodesic flow satisfying the assumptions (II), (III) and (VI) above is non-uniformly hyperbolic with respect to the volume probability measure, and, furthermore, we identified the Oseledets stable and unstable subspaces (cf. the last theorem of this post):

Theorem 2Under the assumptions (II), (III) and (VI) in Theorem 1 above, the geodesic flow isnon-uniformly hyperbolic: more concretely, there exists a subset of full -measure such that the -invariant splitting

into the flow direction and the spaces and ofstable and unstable Jacobi fieldsalong have the property that

for all and .

Today, we want to exploit the non-uniform hyperbolicity of (and the assumptions (I) to (VI) above) in order to deduce the ergodicity of via Hopf’s argument.

For this sake, we organize this post as follows. In the first section, we discuss the geometry of stable and unstable manifolds of : in particular, we will see that these invariant manifolds form *global* laminations with useful *absolute continuity* properties. After that, we describe Hopf’s argument in the second section: from the nice properties of the invariant laminations, we deduce that Birkhoff averages are constant almost everywhere, and, hence, is ergodic. Finally, we conclude this post with a remark (inspired by conversations with Y. Coudène and B. Hasselblatt last November 2013) about the deduction of the *mixing* property for from Hopf’s argument.

**1. Stable manifolds of certain geodesic flows **

** 1.1. Local (Pesin) stable manifolds for certain geodesic flows **

We begin by noticing that a geodesic flow satisfying the assumptions (I) to (VI) of Theorem 1 has “nice” local (Pesin) stable and unstable manifolds through almost every point.

The reader with some experience with non-uniformly hyperbolic systems might think that this is an immediate consequence of the so-called Pesin’s theory. However, this is *not* the case in our setting because the phase space of is *not* assumed to be compact. In other words, we are facing a dilemma: while the non-compactness of is an important point for the applications of Theorem 1 (to moduli spaces equipped with WP metrics), it forbids a naive utilization of Pesin’s theory because of the competition between the dynamical behaviors of in compact regions of and near “infinity” .

Fortunately, Katok and Strelcyn (with the aid of Ledrappier and Przytycki) developed a *generalization* of Pesin’s theory where any “well-behaved” dynamics on non-compact phase space is allowed. Furthermore, Katok-Strelcyn successfully applied their version of Pesin’s theory to the study of dynamical billiards.

Very roughly speaking, Katok-Strelcyn say that if the dynamics of the non-uniformly hyperbolic system “blows up at most polinomially” at infinity , then the hyperbolic (exponential) behavior of is strong enough so that Pesin’s theory can be applied (because is “essentially compact” for practical purposes).

Evidently, this is much easier said than done, and, unfortunately, the discussion of the details of Katok-Strelcyn’s generalization of Pesin’s theory is out of the scope of this post. In particular, we will content ourselves in just mentioning the conditions (I) to (VI) in Theorem 1 were set up by Burns-Masur-Wilkinson in such a way that a geodesic flow satisfying (I) to (VI) also verifies all the requirements to apply Katok-Strelcyn’s work. Here, even though this is philosophically natural, it is worth to point out that the deduction of the conditions to use Katok-Strelcyn’s technology from (I) to (VI) is *far from trivial*: indeed, Burns-Masur-Wilkinson do this after studying (in Appendices A and B of their paper) several properties of Sasaki metric and properties of .

In summary, Burns-Masur-Wilkinson use (I) to (VI) to ensure that Katok-Strelcyn’s generalization of Pesin’s theory applies in the setting of Theorem 1. As a by-product, they deduce the following statement about the existence and absolute continuity of local (Pesin) stable manifolds (cf. Proposition 3.10 of Burns-Masur-Wilkinson paper).

Theorem 3 (“Pesin stable manifold theorem”)Then, there exists a subset of full volume, a \textrm{measurable} function , and a measurable familyLet be the geodesic flow on the unit tangent bundle of a -dimensional Riemannian manifold satisfying the conditions (I) to (VI) of Theorem 1. Denote by the subset of full volume provided by Theorem 2 where is non-uniformly hyperbolic.

of smooth () embedded disks with the following properties. For all :

- , i.e., is tangent to ;
- for all , i.e., is topologically contracted in forward time by ;
- if and only if and , i.e., is local stable manifold (in the sense that it is dynamically characterized as the set of close to whose forward -orbit approaches the forward -orbit of ).

Moreover, the family is absolutely continuous in the sense that the following “Fubini-like statements” hold.

- given a subset of zero volume, one has that the set has zero measure in (with respect to the induced -dimensional Lebesgue measure on ) for almost every ;
- given a -embedded -dimensional open disk and a subset of zero measure (for the induced Lebesgue measure of ), the set
(obtained by saturating by the local stable manifolds passing through it) has zero volume in .

Finally, the analogous assertions about unstable manifolds are also true.

** 1.2. Global stable manifolds of certain geodesic flows **

The Pesin stable and unstable laminations provided by Theorem 3 are *not* sufficient to run Hopf’s argument: as it was explained in the first post of this series, the local stable manifolds could be a priori very *short* (because their radii vary only *measurably* with and so one does not expect for uniform lower bounds on ).

Hence, it is important (for our purposes of using Hopf’s argument) to compare Pesin’s local stable manifolds with *global* objects. Here, the key point is to observe that Theorem 2 says that the tangent space of at is exactly the vector space of *stable Jacobi fields* along the geodesic and, as we will recall in a moment, stable Jacobi fields are naturally related to global objects called *stable horospheres*.

**1.2.1. Stable Jacobi fields and stable horospheres**

Let be a Riemannian manifold. Given an unit tangent generating a geodesic ray such that the sectional curvatures of are negative along and , let us denote by the *stable Jacobi field* associated to : by definition, this is the Jacobi field

where is the Jacobi field satisfying and .

In terms of the description of Jacobi fields via variations of geodesics, the stable Jacobi fields along are obtained by varying through geodesics such that for all (that is, stays always close to in *forward time*). These geodesics are *orthogonal* to a family of immersed hypersurfaces of whose lifts to the universal cover of are the so-called *stable horospheres*.

The stable horospheres can be constructed “by hands” with the aid of Busemann functions as follows.

Let be the quotient of a contractible, negatively curved, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously and suppose that the universal cover of is geodesically convex (i.e., satisfies item (I) of Theorem 1).

In this situation, it is possible to show (see, e.g., Proposition 3.5 in Burns-Masur-Wilkinson paper) that given an unit vector generating an infinite geodesic ray , the functions given by

converge (uniformly on compact sets) as to a convex function

called *stable Busemann function* such that and, for every , the unit vector defines an infinite geodesic ray with

for all . In particular, the geodesics give variations of leading to stable Jacobi fields.

For each , the level set is a connected, complete, codimension submanifold of called *stable horosphere of level *. By definition, the geodesics are orthogonal to the -parameter family of stable horospheres (because stable horospheres are leve sets of and the geodesics point in the direction of the gradient).

The submanifold

of consisting of unit vectors that are orthogonal to the stable horosphere of level is called the *(global) stable manifold* of . This nomenclature is justified by the following property (corresponding to Proposition 3.6 in Burns-Masur-Wilkinson paper). In the context of Theorem 1, suppose that the infinite geodesic ray projecting to a *forward recurrent* geodesic on (i.e., *after* projection to , the unit vector becomes an accumulation point of the set ). Then, for any , the unit vector is tangent to an infinite geodesic ray such that

Furthermore, as . In particular, (stable manifolds are -invariant) and for all (stable manifolds are dynamically characterized by future orbits getting close together).

Remark 1As usual, by reversing the time (via the symmetry ), one can define unstable Jacobi fields, unstable Busemann functions and unstable horospheres.

The following picture (that we already encountered in the last post while discussing Jacobi fields) illustrates the stable and unstable horospheres associated to the vertical geodesic in the hyperbolic plane passing through .

**1.2.2. Geometry of the stable and unstable horospheres**

In this subsection, we make a couple of comments on the geometry of stable and unstable horospheres. More precisely, besides explaining the computation of their second fundamental forms from matrix Riccati equations, we will see that the stable and unstable horospheres are mutually transverse in a quantitative way. Of course, this transversality property of horospheres is another important point in Hopf’s argument (as it allows to control the angle between stable and unstable manifolds).

Let be a geodesic ray such that the sectional curvatures of along are negative. For each , let us denote by the unstable Jacobi field along with (as usual).

Consider the -parameter family of matrices (linear operators) defined by the formula

As we mentioned in this post here, are symmetric, positive-definite operators satisfying the matrix Ricatti equation

(i.e., for all ).

It is possible to show (cf. Eberlein’s survey) that the operator is precisely the second fundamental form at of the unstable horosphere of level .

By reversing the time, we have an analogous operator related to stable horospheres.

Note that, by definition, the stable and unstable subspaces and at an unit vector defining an *infinite* geodesic ray are

In other terms, we have a -invariant splitting

over the set

(where ).

Let us now show that this splitting is locally uniform over .

Proposition 4There exists a continuous function such that the continuous family of conefields

and

meeting only at the origin have the property that

for all .

*Proof:* Our task consists in showing that the functions

of are locally uniformly bounded away from zero.

By symmetry, it suffices to prove that is locally uniformly bounded from below. For the sake of reaching a contradiction, suppose this is not the case. This means that there are sequences , with such that , and .

For each , let be the stable Jacobi fields along induced by , and denote by the (limit) Jacobi field along induced by .

On one hand, for each , the square of the norm of the stable Jacobi field is a decreasing function of . In particular, since , we deduce that is a non-increasing function of .

On the other hand, is a strictly convex function of (because is a perpendicular Jacobi field, cf. Eberlein’s survey).

By putting these two facts together, we see that the function has no critical points. However, . This contradiction proves the desired proposition.

**1.2.3. Absolute continuity of global stable manifolds**

Once we have related Pesin’s stable and unstable manifolds (local objects) to stable and unstable horospheres (global objects), it is not entirely surprising that the absolute continuity properties of Pesin stable manifolds (described in Theorem 3 above) can be “transferred” to horospherical laminations:

Proposition 5Then, there exists a subset of full volume such that the stable Busemann functions are for all . Moreover, the leaves of the stable lamination are -submanifolds of diffeomorphic to . Furthermore, the stable horospherical laminationLet be the geodesic flow on the unit tangent bundle of a -dimensional Riemannian manifold satisfying the conditions (I) to (VI) of Theorem 1. Denote by the subset of the unit tangent bundle of the universal cover of consisting of unit vectors projecting into a forward and backward recurrent geodesic in .

obtained by taking the family of manifolds through the vectors in the projection of to (via ) has the following absolute continuity properties:

- if has zero -volume, then for -almost every and any , the set has zero -dimensional volume in ;
- if is a smooth, embedded, -dimensional open disk and has zero -dimensional volume in , then for any one has where
is the set obtained by saturating with the leaves of the lamination .

Finally, a similar statement holds for the corresponding unstable lamination.

Logically, the statement of this proposition is very close to Theorem 3 about the absolute continuity of Pesin stable manifolds, but the crucial point is that we have now an absolutely continuous stable lamination whose leaves have radii essentially equal to . In other words, the leaves of the stable lamination have a size controlled by the injectivity radius of , a global smooth function, instead of the *a priori* merely measurable function giving the radii of leaves of Pesin’s stable lamination .

The proof of Proposition 5 is not very difficult: it uses the absolute continuity properties of Pesin’s lamination in Theorem 3 and the “contraction of stable horospheres” (i.e., the fact that the forward dynamics of eventually contracts inside ), and it occupies two pages in Burns-Masur-Wilkinson paper (cf. the proof of their Proposition 3.11). However, we will skip this point in favor of discussing Hopf’s argument right now.

**2. Proof of Theorem 1 via Hopf’s argument**

Let be a geodesic flow satisfying the assumptions (I) to (VI) of Theorem 1. We want to show that is ergodic with respect to the volume measure (with normalized total mass).

By Birkhoff’s ergodic theorem, given a continuous function with compact support, the Birkhoff ergodic averages

converge as to the same limit for -almost every .

By definition of ergodicity, our task consists in showing that the function is constant -almost everywhere.

For this sake, let us define the measurable functions

and

Note that, by Birkhoff’s ergodic theorem, there exists a subset of full -measure such that

Moreover, from their definitions, note that the functions , and are -invariant.

The initial observation in Hopf’s argument is the fact that the function , resp. , is *constant* along the stable manifolds , resp. unstable manifolds . In fact, this follows easily from the uniform continuity of the (compactly supported, continuous) function and the fact that as (resp. ) whenever (resp. ).

The basic strategy of Hopf’s argument can be summarized as follows. We want to combine this initial observation with the absolute continuity properties of the stable and unstable horospherical laminations to deduce that is “*locally ergodic*” in the sense that *every* possesses a neighborhood such that the restriction to is -almost everywhere constant.

Of course, since is connected, this local ergodicity property implies that the function is constant -almost everywhere, and, *a fortiori*, is ergodic with respect to . In other terms, our task is reduced to prove the local ergodicity property stated in the previous paragraph.

In this direction, we fix once and for all , we set

and we denote by the -neighborhood of .

Let be the full -volume subset constructed in Proposition 5. For each , we consider the stable leaf , we take its iterates under for , and we saturate the resulting subset with the leaves of the unstable horospherical lamination to obtain the subset

The construction of is illustrated in the figure below: the subset is marked in blue and some leaves of passing through points of are marked in red.

The local ergodicity property stated above is an immediate consequence of the following two claims:

- (a) the restriction of the function to is almost everywhere constant for almost every choice of ;
- (b) is essentially open for almost every near in the sense that there exists a neighborhood of such that has full volume in for almost every choice of .

We establish the first claim (a) by exploiting the initial observation that Birkhoff averages are constant along stable and unstable manifolds and the absolute continuity properties of the stable and unstable horospherical laminations.

More precisely, let us consider again the full volume subset of where (provided by Birkhoff’s ergodic theorem).

By absolute continuity property of (cf. the first item of conclusion of Proposition 5), for almost every , the intersection has full volume in . We affirm that is almost everywhere constant for any such .

In fact, takes a constant value on . Moreover, since on , we also have that takes the constant value on . By combining this fact with the -invariance of , we deduce that takes the constant value on . Furthermore, by putting together this fact with the initial observation that is constant along unstable manifolds , we obtain that takes the constant value on .

Note that, by assumption, is a full volume subset of . Since is a -flow, it follows that is a full volume subset of the -dimensional smooth submanifold . Therefore, from the absolute continuity property of (cf. the second item of conclusion of Proposition 5), we conclude that is a full volume subset of . In particular, we have that takes the constant value on the full volume subset of . Because on , we get that takes the constant value on the full volume subset of , i.e., is almost everywhere constant. This completes the proof of the claim (a).

Remark 2The reader is encouraged to interpret this argument in the light of Figure 2 in order to get a clear picture of the roles of the subsets , and .

We establish now the second claim (b) from the absolute continuity properties of the horospherical laminations and the local uniform transversality of the stable and unstable manifolds.

More concretely, from the absolute continuity property in the first item of the conclusion of Proposition 5, we have that the stable disk , resp. unstable disk , is almost everywhere tangent to the stable direction , resp. unstable direction , for almost every . Since the stable and unstable directions and are contained in the *continuous* families of cones and from Proposition 4, we have that , resp. , is *everywhere* tangent to , resp. for almost every .

In particular, from the -invariance of the stable lamination , we see that the -dimensional disk is everywhere tangent to for almost every . Since the continuous conefields and meet only at the origin (cf. Proposition 4), that is, they are locally uniformly transverse, we conclude that there exists a neighborhood of such that

for almost any . In other words, intersects in a full volume subset. This completes the proof of claim (b).

This concludes our discussion of Hopf’s argument (namely, the derivation of claims (a) and (b)) for the ergodicity of .

Closing this post, let us say a few words about the mixing and Bernoulli properties in the statement of Theorem 1. In Burns-Masur-Wilkinson paper, these properties are deduced from general results of Katok saying that if a *contact* flow is non-uniformly hyperbolic and ergodic, then it is Bernoulli (and, in particular, mixing).

Nevertheless, as it was brought to my attention by B. Hasselblatt and Y. Coudène, the Hopf argument above can be slightly adapted in certain contexts to give mixing and/or mixing of all orders. For example, concerning the mixing property, Y. Coudène, B. Hasselblatt and S. Troubetzkoy showed (in Theorem 3.3) in this recent preprint here that if any -function saturated by stable and unstable sets (in the sense that there is a full measure subset such that whenever and or ) is almost everywhere constant, then the dynamical system is mixing. Also, they have a similar criterion for multiple mixing, and, furthermore, they discuss a couple of non-trivial examples of applications of their criteria.

In the context of Theorem 1, we can deduce the mixing property for from the result of Coudène-Hasselblatt-Troubetzkoy. Indeed, the argument used in the proof of the claim (a) above (during the discussion of Hopf’s argument) also shows that any -function saturated by stable and unstable sets (such as ) is almost everywhere constant, so that Coudène-Hasselblatt-Troubetzkoy mixing criterion “à la Hopf” applies in this setting.

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Of course, there are several ways to come around this little technical subtlety (from the dynamical point of view) in the definition of Kontsevich-Zorich cocycle and this is the main purpose of this post. Evidently, the content of this post is well-known (especially among experts), but I hope that this post will benefit the reader with some background in Dynamical Systems wishing to know the answer to the following question:

*Does the Kontsevich-Zorich cocycle (as it is classically defined) qualifies as a genuine example of linear cocycle in the usual sense in Dynamical Systems?*

**Disclaimer.** Even though this post benefited from my conversations with Jean-Christophe Yoccoz, all errors and mistakes below are my sole responsibility.

**1. The Kontsevich-Zorich cocycle **

The basic references for this section are G. Forni’s paper, J.-C. Yoccoz’s survey and/or this blog post here (where the reader can find some figures illustrating the notions discussed below).

Let be a fixed compact orientable topological surface of genus , let be a non-empty finite set of points of cardinality and let be a list of “ramification indices” such that .

Recall that a *translation surface structure* on is a maximal atlas of charts on such that all changes of coordinates are translation in and, for each , there are neighborhoods , and a ramified cover of degree such that every injective restriction of is a chart of the maximal atlas .

Remark 1Equivalently, we can think of translation structures as the data of a Riemann surface structure on together with an Abelian differential (holomorphic one-form) possessing zeroes of orders at for . However, for the purposes of this post, we will not need this alternative point of view.

Remark 2Since the usual Euclidean area form on is translation invariant, it makes sense to talk about the total area of a translation structure. From now on, we will always implicitly assume that our translation structures are normalized, i.e., they have unit total area. Here, it is worth to point out that this normalization is not important for the definition of Teichmüller and moduli spaces, but it is important for the discussion of the dynamics of the Teichmüller flow on moduli spaces.

We denote by the group of orientation-preserving homeomorphisms of fixing (pointwise), by the connected component in of the identity element, and by the *mapping class group* (sometimes also called modular group).

Note that the group acts (by *pre-composition*) on the set of translation surfaces: given and a translation surface structure on , we get a translation structure by defining .

In this setting, the *Teichmüller space* is the quotient of the set of translation structures on by the action of and the *moduli space* is the quotient of the set of translation structures on by the action of . By definition, the moduli space is the quotient of Teichmüller space by the action of the mapping class group .

Remark 3The Teichmüller space is a manifold, but the moduli space is an orbifold (not a manifold) in general. We will come back to this point later in this post.

The group acts (by *post-composition*) on the Teichmüller space : given and a translation structure , we define (note that this action is well-defined because the conjugation of a translation in by the linear action of the matrix is still a translation). Furthermore, since acts by pre-composition and acts by post-composition, these actions commute and, hence, the action of descends to the moduli space .

The actions of the diagonal subgroup of on Teichmüller and moduli spaces are called *Teichmüller flow*.

The dynamics of the Teichmüller flow and/or -action on moduli spaces of (normalized) translation surfaces is a rich subject with interesting applications to the Ergodic Theory of some parabolic systems (such as interval exchange transformations and billiards in rational tables): see, for example, these posts here and here for more details.

A main character in the investigation of the -action on moduli spaces of (normalized) translation surfaces is the so-called *Kontsevich-Zorich cocycle*. Very roughly speaking, this cocycle was introduced by Kontsevich and Zorich as a practical way to extract the “interesting part” of the derivative cocycle of the Teichmüller flow.

Formally, the Kontsevich-Zorich (KZ) cocycle is usually defined as follows (compare with Forni’s paper). Let be the *vector bundle* over Teichmüller space whose fibers are the absolute homology group with real coefficients. One usually refers to as the *Hodge bundle* over .

Remark 4The reader with some background in Complex Geometry might have thought that this notion is very similar to the Hodge bundle over Teichmüller and moduli spaces of algebraic curves (Riemann surfaces) obtained by attaching the space of holomorphic -forms to a Riemann surface .In fact, this is no coincidence and the nomenclature “Hodge bundle” for is a “popular” abuse of notation in the literature about the Teichmüller flow. In fact, this abuse of notation goes beyond this: one could also construct (trivial) bundles over Teichmüller spaces by taking the fibers to be the absolute homology group with complex coefficients or the absolute cohomology group or with real or complex coefficients. These variants are closely related to each other (because and the first absolute homology and cohomology groups of a surface are dual [by Poincaré duality]) and they are also called Hodge bundle in the literature (depending on the author’s taste).

The vector bundle is *well-defined* and *trivial*, i.e., : in a nutshell, this is a consequence of the fact that a homeomorphism that is isotopic to the identity (such as the elements of ) act trivially on homology.

By taking the quotient of by the natural action of the mapping class group on *both factors*, we get the so-called *Hodge bundle*

over the moduli space .

In this context, the (trivial) cocycle

over the Teichmüller flow on Teichmüller space given by

for and descends to the so-called *Kontsevich-Zorich cocycle* on the Hodge bundle over moduli space (by taking the quotient by the action of ). Here, it is worth to observe that the Kontsevich-Zorich cocycle is well-defined (i.e., we can take this quotient) because of the fact that acts by pre-composition and acts by post-composition on Teichmüller spaces (so that these actions commute).

Remark 5The Kontsevich-Zorich cocycle could also be defined more generally by taking the quotient of the trivial cocycle over the action of full group (and not only ) on Teichmüller space.

**2. Is the KZ cocycle a linear cocycle? **

The reader with some familiarity with Dynamical Systems might have noticed some similarities between the notions of Kontsevich-Zorich cocycle and a linear cocycle over (discrete or continuous time) dynamical system.

In fact, let us recall that a *linear cocycle* , , over a flow , , is a flow on a vector bundle such that (i.e., projects onto ) and is a vector bundle automorphism, i.e., for all , the restriction of to is a linear map from the fiber on the fiber .

Example 1The trivial cocycle on the trivial bundle over a flow is . In particular, the cocycle on defined above is an example of trivial cocycle.

Example 2The derivative map of a smooth flow on a smooth manifold is an important class of examples of linear cocycles.

Given that the Kontsevich-Zorich cocycle on moduli spaces projects to the Teichmüller flow on moduli spaces and it acts on the fibers of via the (symplectic) action on homology of the elements of the mapping class group , one might be tempted to qualify the Kontsevich-Zorich cocycle as a linear cocycle.

*However*, a closer inspection of the definitions reveals that:

The Kontsevich-Zorich cocycle is not *always* a linear cocycle!

Actually, the fact that KZ cocycle is not a linear cocycle in general is *not* its fault: in order to talk about *linear* cocycles one needs *vector bundles*, and, as it turns out, the fibers of the Hodge bundle over moduli space are *not* vector spaces over the orbifold points of moduli spaces.

More precisely, we see from the definition that the fiber of at a translation surface is the quotient where is the group of automorphisms of , that is, the group of homeomorphisms of fixing pointwise whose local expressions in the charts are translations of .

Note that is a finite group: for instance, any element of is holomorphic (with respect to the Riemann surface structure underlying ) and, hence, by Hurwitz’s automorphisms theorem, we have that has cardinality . (Actually, even though Hurwitz’s theorem is sharp, this estimate of is not optimal: see, e.g., this paper of Schlage-Puchta and Weitze-Schmithuesen)

Therefore, the fiber of at is not very far from a vector space: it differs from by the quotient by (the action on homology of) the finite group (of “symplectic matrices”).

Nevertheless, when the translation surface is an *orbifold* point of moduli space (i.e., when is non-trivial), the fiber is not necessarily a vector space. (A simple concrete example of such a situation is the cone obtained from the quotient of by the finite group generated by the rotation of angle )

In summary, KZ cocycle is not always a linear cocycle because the Hodge bundle over moduli space is not always a vector bundle.

In other terms, the moduli space is an orbifold (but not a manifold in general), the Hodge bundle is an *orbifold vector bundle* (in general) and, thus, KZ cocycle is an *orbifold linear cocycle* (in general).

Example 3A concrete description of the Eierlegende Wollmilchsau is the following. We consider the quaternion group , we take an unit square in for each , and we glue (by translation) the vertical rightmost side of to the vertical leftmost side of and we glue (by translation) the horizontal top side of to the horizontal bottom side of . In this way, one obtais a translation surface where has genus , consists of four points and .One of my favorite examples of translation surface with a non-trivial group of automorphisms is the so-calledEierlegende Wollmilchsau.

A simple argument (see, e.g., this paper here) shows that the group of automorphisms of the Eierlegende Wollmilchsau is isomorphic to the quaternion group .

Example 4Some moduli spaces are manifolds and the corresponding Hodge bundles are vector bundles.For instance, the so-called minimal stratum of translation surfaces on a genus surface with a single marked point is a manifold because it can be shown (see, e.g., Proposition 2.4 in this paper here) that the automorphism group of any translation surface is trivial.

**3. Dynamics of the KZ cocycle? **

From the point of view of Topology and Algebraic Geometry, the “orbifoldic” nature of KZ cocycle is not surprising. Indeed, this kind of object is very common when studying monodromy representations and, also, one can overcome the “orbifoldic” nature of KZ cocycle by taking *covers* of moduli spaces in order to “kill” orbifold points. In particular, an orbifold linear cocycle is as good as a linear cocycle for topological considerations.

On the other hand, for *dynamical* considerations, the classical definition of KZ cocycle as an orbifold linear cocycle deserves further discussion.

For example, given an ergodic Teichmüller flow invariant probability measure on , it is desirable to apply Oseledets theorem to KZ cocycle in order to talk about its Lyapunov exponents and/or Oseledets subspaces. However, the Oseledets theorem deals only with linear cocycles and, thus, the fact that the KZ cocycle is merely an orbifold linear cocycle, or, more precisely, the fibers of Hodge bundle are not vector spaces, imposes some technical difficulties.

Remark 6For ergodic-theoretical purposes, the technical point pointed out above only shows up when -almost every translation surface is an orbifold point.In particular, the discussion in this section does not concern the so-calledMasur-Veech probability measureson moduli spaces (because its generic points have trivial group of automorphisms). This explains why the orbifoldic nature of KZ cocycle is never discussed in earlier papers in the literature (such as Forni’s paper): in those paper, the authors were concerned exclusively with the behavior of almost every trajectory with respect to Masur-Veech measures.

Remark 7The orbifoldic nature is discussed (in an implicit way at least) in the literature on Veech surfaces. Recall that a Veech surface is a translation surface whose -orbit in moduli space is closed. As it turns out, the stabilizer in of a Veech surface is a lattice, so that its -orbit is naturally isomorphic to the unit cotangent bundle of the finite area hyperbolic surface . If the Veech surface has a non-trivial group of automorphisms, the Hodge bundle over its -orbit (and hence the corresponding KZ cocycle) is orbifoldic, but one can get around this by studying the group of so-calledaffine diffeomorphisms, a sort of “finite cover” of in view of a natural exact sequence . See, e.g., this survey paper of P. Hubert and T. Schmidt (or our joint paper with J.-C.Yoccoz).

Fortunately, for the sake of the *definition* of Lyapunov exponents of KZ cocycle with respect to an ergodic Teichmüller flow invariant probability measure , the possible ambiguity coming from the fact that the KZ cocycle is a “linear cocycle up to ” is *irrelevant* (cf. Section 4.3 of our paper with J.-C. Yoccoz and D. Zmiaikou). In fact, the Lyapunov exponents are defined by measuring the exponential rate of growth

of vectors (along typical trajectories), and the ambiguity caused by the fact that is a well-defined linear operator only up to the matrices in (action on homology of) does not change these rates because is a *finite* group and, hence, the possible values of (after composing with the elements of ) are uniformly related to each other by universal multiplicative constants (whose effects disappear when considering the expression ). In other terms, the Lyapunov exponents of orbifold linear cocycles *are* well-defined!

Unfortunately, there is no “cheap” solution (similar to the previous paragraph) for the definition of *Oseledets subspaces* of KZ cocycle: one needs linear structures on the fibers of the Hodge bundle to talk about them. Logically, this is an annoying situation because Oseledets subspaces are useful: for example, the analysis of these subspaces plays a major role in the recent breakthrough paper of A. Eskin and M. Mirzakhani about Ratner-like theorems for the -action on moduli spaces.

As it was already mentioned in the beginning of this section, the way out of this dilemma is to pick a cover of moduli space where all orbifold points disappear and to lift the Hodge bundle to this cover.

Of course, there are *several ways* of picking such a cover, but the whole point of this post is that certain covers are better than others depending on our purposes.

For example, the Teichmüller space is a cover of having no orbifold points (because an automorphism of a translation surface of genus that is isotopic to the identity is the identity), and the Hodge bundle is a vector bundle. Nevertheless, the Teichmüller space of (normalized) translation structures is not *dynamically* interesting: for example, there are no *finite* Teichmüller invariant measure (and, thus, we can not use the standard tools from Ergodic Theory). This situation is very similar to the case of geodesic flows on the unit cotangent bundle of a finite-area hyperbolic surfaces (if we think of as moduli space, as Teichmüller space and the geodesic flow as Teichmüller flow): even though there are plenty of finite geodesic flow invariant measures on , there are no finite geodesic flow invariant measures on the cover (and, in fact, the dynamics of the geodesic flow on unit cotangent of the hyperbolic plane is rather boring). In summary, despite the fact that the Hodge bundle is an honest vector bundle over Teichmüller space , we can not use it to define Oseledets subspaces (or, in general, do non-trivial dynamics) because the Teichmüller flow on is not dynamically rich.

The “failure” (from the dynamical point of view) in the previous paragraph suggests that we should try picking *finite* covers of moduli spaces (having no orbifold points). Indeed, the lift of Teichmüller flow invariant probability measures leads to a finite measures in that case and we are in good position to discuss Ergodic Theory.

In this direction, J.-C. Yoccoz, D. Zmiaikou and myself considered the cover of obtained by marking an horizontal separatrix issued of a conical singularity of the translation surface. This is a finite cover because there are exactly outgoing horizontal separatrices at a conical singularity with total angle . Furthermore, has no orbifold points: indeed, any automorphism of a translation surface that fixes an horizontal separatrix issued of a conical singularity is the identity. Moreover, the diagonal subgroup (Teichmüller flow) still acts on because the matrices respect the horizontal direction (and, thus, horizontal separatrices).

In particular, we can talk about the Oseledets subspaces of the KZ cocycle over Teichmüller flow at the level of the lift of the Hodge bundle to (because this “lifted KZ cocycle” over Teichmüller flow is a genuine linear cocycle over a probability measure preserving flow and hence we can apply Oseledets theorem).

For the purposes of our joint paper with J.-C. Yoccoz and D. Zmiaikou, the lift of the KZ cocycle to the Hodge bundle over the finite cover of moduli space was adequate (and this is why we decided to stick to it in our paper).

However, we should confess that we were not completely happy with because does *not* act on (even though its diagonal subgroup do act!). Therefore, the more general version of the KZ cocycle over the -action on moduli spaces in Remark 5 above (also very important for the applications of the Teichmüller flow) can *not* be lifted to the Hodge bundle over .

Actually, the reason why does not act on is very simple: the action of the rotation subgroup where is ill-defined. In order to see this, let us consider the following translation surface of genus with a marked (in blue) horizontal separatrix issued of its unique conical singularity :

Let us try to define the action of on by letting vary from to . Starting from , let us slowly apply the rotation matrices to the translation surface for small positive values of :

In this way, we get a new translation surface and the horizontal separatrix is sent into a *non-horizontal separatrix* . Thus, and, *a fortiori*, the natural “reflex” of posing fails.

Evidently, for any small positive angle , the non-horizontal separatrix is very close to the horizontal separatrix (obtained by “rotating” by an angle inside ) indicated below

If we do this, then, by letting vary from to , we would be force to impose where is the “previous” horizontal separatrix issued from the singularity in the “natural cyclic order” (obtained by rotating around the singularity) indicated (in red) below However, since the horizontal separatrices and are *distinct*, we have that and are *distinct* points of , a contradiction with the fact that the rotation matrix must act by the identity map on .

A simple inspection of the argument above shows that the *finite* cover of (obtained by “replacing” the rotation (circle) group by its -fold cover , but keeping the “same” diagonal subgroup ) does act on ! In other words, “almost acts” on . Nevertheless, since the (non-trivial) finite cover is *not* an algebraic group (unlike itself), the natural action on the Hodge bundle over is not so useful from the point of view of Dynamical Systems.

In summary, despite the usefulness of for the study of the KZ cocycle over the Teichmüller flow on moduli spaces, it is desirable to find an alternative finite cover of moduli spaces having no orbifold points where still acts.

One solution to this problem is to take the quotient of Teichmüller space by a *torsion-free* finite index subgroup of : indeed, is a finite cover of moduli space because has finite index, has no orbifoldic points because is torsion-free and acts on because acts by *post-composition* and acts by *pre-composition* on .

Here, a result of J.-P. Serre (see also the book of B. Farb and D. Margalit for a “modern” exposition of Serre’s result [in the context of moduli spaces of Riemann surfaces]) produces many of those subgroups with the desired properties: in fact, given any integer , Serre showed that the subgroup

consisting of elements of the mapping class group acting trivially on the homology of with coefficients in is torsion-free. (This finite cover of moduli space was already mentionned in this blog: see this post here)

In summary,

The lift of the KZ cocycle to the Hodge bundle over defines a linear cocycle over the -action on moduli spaces that we could (should?) call KZ cocycle in *dynamical* discussions (where certain specific notions such as Oseledets subspaces are needed).

Remark 8For the sake of comparison, the finite cover might be somewhat big relative to . In fact, is a cover of degree in general, while, from the fact that the action on homology of the mapping class group surjects into , we have that in general is a cover of degree (a quantity of the form for prime).

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The plan for this post is the following. After quickly reviewing in Section 1 below some basic features of the geometry of tangent bundles of Riemannian manifolds, we will estimate the first derivative of geodesic flows on certain negatively curved manifolds in terms its sectional curvatures (as promised last time). Finally, we will complete today’s discussion by proving the first part of Burns-Masur-Wilkinson ergodicity criterion (i.e., we will show that any geodesic flow verifying the assumptions of Burns-Masur-Wilkinson is non-uniformly hyperbolic in the sense of Pesin’s theory), while leaving the second part of Burns-Masur-Wilkinson ergodicity criterion (i.e., the verification of ergodicity via Hopf’s argument) for the next post of this series.

**1. Geometry of tangent bundles **

** 1.1. Riemannian metrics, Levi-Civitta connections and Riemannian curvature tensors **

Let be a Riemmanian manifold and denote by its Riemannian metric of .

Let be the associated Levi-Civita connection, i.e., the unique connection (“notion of parallel transport”) that is symmetric and compatible with the Riemannian metric . Given a curve on , the covariant derivative along is

(and it should *not* be confused with ). Sometimes we will also denote the covariant derivative simply by when the curve is implicitly specified: for example, given a vector field along a curve (of footprints), we write where is an extension of to .

In this setting, recall that a curve is a geodesic if and only if for all .

Since the equation is a first order ODE (in the variables ), we have that geodesics are determined by the initial vector . Furthermore, any geodesic has constant speed, i.e., the quantity measuring the square of size (norm) of the tangent vector is constant along : in fact, using the compatibility between and , one gets

for all .

The lack of commutativity of the Levi-Civitta connection is measured by the Riemannian curvature tensor

In terms of the Riemannian curvature tensor , the sectional curvature of a -plane spanned by two vectors and is

** 1.2. The tangent bundle to a tangent bundle **

The tangent bundle of the tangent bundle of is a bundle over in three natural ways:

- (a) where is the natural projection;
- (b) where is the natural projection;
- (c) where is defined as follows: given tangent at to a curve , we set where is the curve of footprints of the vectors ;

In this context, the vertical, resp. horizontal, subbundle of is , resp. . The vertical, resp. horizontal, subbundle is naturally identified with via , resp. . The vertical subbundle is transverse to the horizontal subbundle and the fiber of at can be identified via the map .

Geometrically, the roles of the vertical and horizontal subbundles are easier to understand in the following way. Given an element of tangent to a curve with , let be the curve of footprints of in . In this setting, the identification of with a pair of vectors via the horizontal and vertical subbundles simply amounts to take

In other terms, the component of in the horizontal subbundle measures how fast is moving in while the component of in the vertical subbundle measure how fast is moving in the fibers of .

This way of thinking as a bundle over leads to the following natural Riemannian metric on : given , we define

This metric is called *Sasaki metric* and the geometry of with respect to this Riemannian metric will be useful in our study of geodesic flows.

Remark 1Sasaki metric is induced by the symplectic form

in the sense that

where . The symplectic form is the pullback of the canonical symplectic form on the cotangent bundle by the map associating to the linear functional .

For the reader’s convenience, let us mention the following three useful facts about Sasaki metric:

- Sasaki showed that the fibers of the tangent bundle are totally geodesic submanifolds of equipped with Sasaki metric;
- A parallel vector field on viewed as a curve on is a geodesic for Sasaki metric that is always orthogonal to the fibers of ;
- by Topogonov comparison theorem, for close to , one has
where is the vector obtained by parallel transporting along the geodesic connecting to and is the distance associated to Sasaki metric; here, how close must be from depends only on the sectional curvatures of Sasaki metric in a neighborhood of ;

**2. First derivative of geodesic flows and Jacobi fields **

** 2.1. Computation of the first derivative of geodesic flows **

Let be the geodesic flow associated to a Riemannian manifold . By definition, given a tangent vector , we define where is the unique geodesic of with . Here, it is worth to point out that the geodesic flow is always locally well-defined but it might be globally ill-defined. Moreover, the geodesic flow preserves the Liouville measure (i.e., the volume form on induced by Riemannian metric of ).

We want to describe and, from the definition of first derivative, this amounts to study (-parameter) *variations of geodesics*.

More precisely, let be a (smooth) map such that, for each , is a geodesic of . Intuitively, is a one-parameter variation of the geodesic .

Define the vector field along the geodesic . It is well-known that satisfies the Jacobi equation

where is the covariant derivative (along ) and is the Riemannian curvature tensor. In other terms, is a *Jacobi field*, i.e., a vector field satisfying Jacobi’s equation.

Observe that Jacobi’s equation is a second order linear ODE. In particular, a Jacobi field is determined by the initial data .

The pair corresponds to the tangent vector at to the curve in (under the identification described above [in terms vertical and horizontal subbundles]). Indeed, the curve of footprints of is , so that the tangent vector at of is represented by

Here, the symmetry of the Levi-Civitta connection was used.

Similarly, the pair represents the tangent vector at to the curve . Therefore, represents

In summary, Jacobi fields are intimately related to the first derivatives of geodesic flows:

Proposition 1The image of the tangent vector under the derivative of the geodesic flow is the tangent vector where is the (unique) Jacobi field with initial data along the (unique) geodesic with .

** 2.2. Perpendicular Jacobi fields and invariant subbundles **

A concrete example of Jacobi field along a geodesic is : indeed, in this context, and , so that Jacobi’s equation is trivially verified. Geometrically, this Jacobi field correspond to a trivial variation of the geodesic where the initial point moves along and/or the speed of the parametrization of changes, i.e., .

In general, a Jacobi field along a geodesic that is tangent to has the form for some : in fact, for with , one has , so that Jacobi’s equation reduces to , i.e., for all .

Hence, a Jacobi field along a geodesic is interesting only when it is not completely tangent to the geodesic, or, equivalently, when it has some non-trivial component in the perpendicular direction to the geodesic.

A Jacobi field along a geodesic has the following geometrical properties:

- the component of makes constant angle with , i.e., the quantity is constant;
- if both components of are orthogonal to at some point, then they stay orthogonal all along , i.e., if and for some , then for all ;

We say that a Jacobi field along a geodesic is a *perpendicular Jacobi field* whenever both components of are orthogonal to .

From the properties of Jacobi fields discussed above, we see that any Jacobi field along a geodesic has a decomposition

where is a perpendicular Jacobi field and is a Jacobi field tangent to .

After this little digression on Jacobi fields, let us use them to introduce relevant invariant subbundles under the first derivative of a geodesic flow .

We begin by recalling that the norm of a tangent vector stays constant along its -orbit, i.e., preserves the energy hypersurfaces (for each ). In particular, the first derivative of the geodesic flow preserves the tangent bundle (to the unit tangent bundle of M).

We affirm that, under the identification for , the fiber (of the subbundle of ) corresponds to the set of pairs with .

In fact, note that an element of is tangent at to a variation of geodesics parametrized by arc-length, i.e.,

for all , such that the geodesic satisfies and the Jacobi field corresponding to verifies .

The desired property now follows from the following calculation:

The invariant subbundle itself admits a decomposition into two invariant subbundles, namely,

where is the vector field generating the geodesic flow and is the orthogonal complement of . In fact, under the identification for , the vector is and the elements of have the form with , . In particular, the -invariance of follows from the fact (mentioned above) that a Jacobi field satisfying and for some is a perpendicular Jacobi field (i.e., and for all ).

In summary, the action of on has two complementary invariant subbundles, namely, the span of the vector field generating the geodesic flow and its orthogonal consisting of perpendicular Jacobi fields. Since acts isometrically in the direction of , our task is reduced to study the action of on perpendicular Jacobi fields.

** 2.3. Matrix Jacobi and Ricatti equations **

We want to describe the matrix of acting on the vector space of perpendicular Jacobi fields. For this sake, let be an orthonormal basis for the tangent space of , and denote by the parallel transport of this orthonormal basis along the geodesic .

Define the matrix whose entries are

where is the Riemannian curvature tensor.

Note that any Jacobi field along can be written as . In this setting, Jacobi equation becomes

and, as usual, a solution is determined by the values and .

We can write solutions of the Jacobi equation above in a practical way by considering a matrix solution of the matrix Jacobi equation:

If is non-singular, the matrix

satisfies the matrix Ricatti equation

Remark 2The matrix is symmetric if and only if one has

for any two columns and of . Here, is the standard symplectic form of .

** 2.4. An estimate for the first derivative of a geodesic flow **

After these preliminaries on the geometry of tangent bundles, geodesic flows and Jacobi fields, we are ready to prove the following result stated as Theorem 11 in our previous post (but whose proof was postponed for this post).

Theorem 2Let be a negatively curved manifold. Let and consider a geodesic. Suppose that is a Lipschitz function such that, for each , the sectional curvature of any plane containing is greater than or equal to , and denote by the solution of Ricatti’s equation

with initial data . Then, the first derivative of the geodesic flow at time satisfies the estimate

From our discussion so far, the task of estimating the norm is equivalent to provide bounds for in terms of where is a perpendicular Jacobi field along (cf. Proposition 1 and Subsection 2).

We begin by estimating these quantities for two special subclasses of perpendicular Jacobi fields defined as follows. Let and be the (fundamental) solutions of the matrix Jacobi equation

with initial data and . Note that, by definition, all Jacobi fields with , resp. all Jacobi fields with , have the form , resp. , i.e., they are obtained by applying the matrices , resp. , to a vector , resp. . In this setting, the “other” component , resp. (of the Jacobi field , resp. , viewed as a tangent vector to ) can be recovered by applying the matrix , resp. , to , resp. .

Remark 3Very roughly speaking, the idea behind the choice of the subclasses and is that are Jacobi fields belonging to a certain “stable cone” and are Jacobi fields belonging to a certain “unstable cone” (compare with the discussion in the next Section).

Our first lemma says that the tangent vectors associated to Jacobi fields as above do *not* growth in forward time.

Lemma 3Let be a perpendicular Jacobi field along such that . Then,

In particular,

*Proof:* One of the consequences of negative sectional curvatures along is the fact that the functions and are strictly convex for any perpendicular Jacobi field (see, e.g., Eberlein’s survey).

In our context, this implies that is a (strictly) convex function decreasing from to in the interval . Therefore,

Since and for close to (because ), we deduce that

This completes the proof of the lemma.

Our second lemma says that the the growth in forward time of tangent vectors associated to Jacobi fields as above is reasonably controlled in terms of the solution of Ricatti’s equation with (where is the Lipschitz function controlling some sectional curvatures of ).

Lemma 4Let be a perpendicular Jacobi field along with . Then,

*Proof:* By definition, . Thus, and, *a fortiori*,

On the other hand, since , we see that and, hence,

These inequalities show that the proof of the lemma is complete once we can prove that for all .

In this direction, let us observe that the matrix is symmetric because it verifies (in a trivial way) the condition of Remark 2. Therefore, the norm of is given by the expression

where ranges from all unit vectors. In particular, our task is reduced to show that

for all unit vectors , where .

From the matrix Ricatti equation, we see that

Since the Lipschitz function controls the sectional curvatures (of planes containing ) along and the matrix is symmetric, we can estimate the right-hand side of the previous inequality as

On the other hand, since is a unit vector, the Cauchy-Schwarz inequality implies that . Therefore, the right-hand side of the previous inequality is bounded by

From this differential inequality and the facts that and , we can easily deduce that for all from the standard continuity argument.

Finally, we can complete the proof of the lemma by observing that the symmetric matrix is positive definite for : this follows from the facts that and satisfies matrix Ricatti equation associated to a negatively curved manifold (cf. Eberlein’s book). Therefore, and for all and all unit vector , so that

as desired.

Once we know how to control the growth of and for Jacobi fields and as above, the idea to estimate the growth of for an arbitrary perpendicular Jacobi field (thus completing the proof of Theorem 2) is to produce a decomposition of the form

where and the norms of and are controlled in terms of the norm of .

For this sake, define

and we set and .

First, note that the vector is well-defined, i.e., the matrix is invertible. Indeed, we already saw that the matrices and are symmetric (because they satisfy (in a trivial way) the condition of Remark 2) and that the matrix is positive definite (because satisfies matrix Ricatti equation, the manifold is negatively curved and imply that is positive-definite for (cf. Eberlein’s book). Furthermore, all eigenvalues of the matrix are : in fact, any eigenvalue of has the form for some unit vector , and because is a convex function (see, e.g., Eberlein’s survey) decreasing from to in the interval (with ). Therefore, the matrix is a symmetric matrix whose eigenvalues are and, hence, is an invertible matrix satisfying

Secondly, we claim that the Jacobi fields and give the desired decomposition. In fact, since , and are Jacobi fields, our claim follows from the facts that and .

Finally, let us estimate the (Sasaki) norms of and in terms of . We begin by observing that it suffices to estimate the Sasaki norm of because

On the other hand, the (Sasaki) norm of is not difficult to bound:

Since (cf. the proof of Lemma 4) and , we can estimate the right-hand side of the previous inequality by

By putting together these estimates of the Sasaki norms of and and Lemmas 3 and 4, we deduce that

This completes the proof of Theorem 2.

**3. Hyperbolicity of geodesic flows on certain negatively curved manifolds **

In this section, we will partly fulfill our promise in our previous post by giving the first steps towards the proof of Burns-Masur-Wilkinson ergodicity criterion:

Theorem 5 (Burns-Masur-Wilkinson)Suppose that:Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously. Denote by the metric completion of and the boundary of .

- (I) the universal cover of is
geodesically convex, i.e., for every , there exists an unique geodesic segmentinconnecting and .- (II) the metric completion of is
compact.- (III) the boundary is
volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfiesfor every .

- (IV) has
polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial boundfor every .

- (V) has
polynomially controlled injectivity radius, i.e., there are constants and such thatfor every (where denotes the injectivity radius at ).

- (VI) The
first derivative of the geodesic flowispolynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :Then, the Liouville (volume) measure of is finite, the geodesic flow on the unit cotangent bundle of is defined at -almost every point for all time , and the geodesic flow is

non-uniformly hyperbolic(in the sense of Pesin’s theory) andergodic.

Actually, the geodesic flow is Bernoulli and, furthermore, its metric entropy is positive, finite and is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of counted with multiplicities).

More precisely, our plan for the rest of this post is to show the non-uniform hyperbolicity of the geodesic flow described in the statement above. Then, we will leave the proof of the ergodicity of (via Hopf’s argument) for the next post of this series.

We start by noticing that has finite -volume: this is an easy consequence of the compactness of (assumption (II)) and the volumetrically cusp-like assumption (III) on .

Next, let us check that the geodesic flow in the statement of Burns-Masur-Wilkinson ergodicity criterion is defined for all time for almost every initial data . For this sake, denote by the natural projection and set

and

By definition,

and, *a fortiori*,

In particular, since the geodesic flow preserves the measure , our task of showing that is defined for all time for almost every initial data is reduced to prove that has zero -measure.

In order to compute the -measure of , let us estimate the -measure of for each along the following lines. Note that

where consists into the unit tangent vectors flowing into for some time between and . By definition, , so that

where . Here, we used the fact that is -invariant (for the first inequality) and the assumption (III) (for the second inequality). It follows that

for all . Hence, has zero -measure and is defined for all time for almost all initial data.

Remark 4The reader certainly noticed that we do not the full strength of assumption (III) to deduce the long-term existence of at almost every point: in fact, the weaker condition works equally well. Nevertheless, we will see below that the full strength of assumption (III) is helpful to ensure the existence of Lyapunov exponents for the geodesic flow .

Now, let us show that the geodesic flow is non-uniformly hyperbolic in the sense of Pesin theory, i.e., all (transverse) Lyapunov exponents are non-zero.

We start by verifying that the Lyapunov exponents of are well-defined (at almost every point): by Oseledets multiplicative ergodic theorem, it suffices to check the -integrability of the derivative cocycles and associated to the time- and time-maps and , that is,

By symmetry (or reversibility of the geodesic flow), we have to consider only the -integrability of . We estimate the integral above for by noticing that

Since is compact (by assumption (II)), we need to show only that the series above is convergent and this is not hard to see: on one hand, we already saw that for some (as a consequence of assumption (III), and, on the other hand, on by assumption (VI), so that

By Oseledets theorem, once we know the -integrability of the derivative cocycle, we have that, for almost every , there are real numbers

called Lyapunov exponents and a -invariant splitting

into Lyapunov subspaces such that, for every ,

In the context of a geodesic flow , recall that the derivative cocycle preserves the decomposition , and acts isometrically along and preserves . This implies that the Lyapunov exponent of along is zero and the derivative cocycle has Lyapunov exponents counted with multiplicity (i.e., we count -times the Lyapunov exponent ) along .

Remark 5In fact, the derivative cocycle preserves a natural symplectic form on . In particular, the Lyapunov exponents are organized in a symmetric way around the origin in the sense that is a Lyapunov exponent whenever is a Lyapunov exponent.

By definition, is called *non-uniformly hyperbolic* whenever all Lyapunov exponents along (sometimes called transverse Lyapunov exponents) are non-zero.

In our context (of the statement of Burns-Masur-Wilkinson ergodicity criterion), we will prove the non-uniform hyperbolicity of by exploiting the negative curvature of . More concretely, the negative curvature of implies that:

- for any non-trivial perpendicular Jacobi field , the functions and are strictly convex (thanks to Jacobi’s equation);
- for each geodesic ray and for each , there exists an unique perpendicular Jacobi field along with such that
for all .

See, e.g., Eberlein’s book for more explanations. In the literature, is called an *unstable* Jacobi field and it is usually constructed as the limit where is the Jacobi field with and . Similarly, we can define *stable* Jacobi fields along geodesic rays by reversing the time of the geodesic flow. The Figure 2 above illustrates stable (“blue”) and unstable (“red”) Jacobi fields along a vertical geodesic in the hyperbolic plane.

We will discuss stable and unstable Jacobi fields in more details in the next post of this series (because they describe the stable and unstable manifolds of and Hopf’s argument depend crucially on the features of stable and unstable manifolds). For now, we just need to know that, if is negatively curved and is defined for all time at , then

where ,

and

In other terms, where and are -dimensional subspaces related to stable and unstable Jacobi fields. See, e.g., Eberlein’s book for a proof of this fact.

In this setting, the non-uniform hyperbolicity of is a direct consequence of the following lemma relating stable and unstable Jacobi fields to Lyapunov subspaces:

Lemma 6There exists a -invariant subset of full -measure such that

*Proof:* Denote by the set of unit vectors such that:

- is defined for all time ;
- the Lyapunov exponents and Lyapunov subspaces are defined for ;
- is
*uniformly recurrent*under in the sense that, for any neighborhood of , there exists such that the sets have Lebesgue measure for all sufficiently large.

Note that is -invariant and it has full -measure: our previous discussion showed that the first two conditions hold for almost every and the third condition holds in a full measure subset thanks to Birkhoff’s ergodic theorem.

We affirm that satisfies the conclusions of the lemma. In fact, by the reversibility of the geodesic flow , it suffices to show that

for all .

For this sake, given , we fix a neighborhood of and a real number such that if is an unstable Jacobi field along a geodesic with , then

The choice of and is possible because is negatively curved and is an increasing strictly convex function whose second derivative is controlled by Jacobi’s equation.

Since is uniformly recurrent for , we have that

for all . Because , we know that has Lebesgue measure for some and for all sufficiently large. Therefore, for any unstable Jacobi field along , one has

for all sufficiently large. It follows from the definitions that

for any , and, hence,

Similarly, . Because , these inclusions must be equalities and the proof of the lemma is complete.

For later reference, we summarize the results of this section in the following statement:

Theorem 7Under the assumptions (I) to (VI) in Theorem 5 above, the geodesic flow is non-uniformly hyperbolic: more concretely, there exists a subset of full -measure such that the -invariant splitting

into the flow direction and the spaces and of stable and unstable Jacobi fields along have the property that

for all and .

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- a) Vadim Kaloshin and Yuri Lima are organizing a “Summer School on Dynamical Systems” to be held at the University of Maryland from August 17th to 25th, 2014. There will be four minicourses (whose titles are available here) by Dmitry Dolgopyat, Giovanni Forni, Anton Gorodetski and Vadim Kaloshin.
- b) Pascal Hubert, Erwan Lanneau and Anton Zorich are organizing the workshop “Dynamics and Geometry in Teichmueller Spaces” to be held at CIRM, Luminy/Marseille, France, next year (from July 6th to 10th, 2015 to be precise).

The first event is aimed at graduate students interested in learning some recent topics in Dynamics and also undergraduate students with some background in Dynamical Systems wishing to pursue her/his studies in Dynamics. The details for this event are being uploaded at the summer school webpage and the organizers (Vadim Kaloshin and Yuri Lima) will be happy to provide extra information for all potential participants.

The second event is a research conference around Teichmueller and moduli spaces from both the geometrical and dynamical points of view. The details for this conference are still being defined (as far as I know) and one is encouraged to write to the organizers (Pascal Hubert, Erwan Lanneau and/or Anton Zorich) for more informations.

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This was the second talk of a new “flat surfaces” seminar organised by himself, Anton Zorich and myself at Instut Henri Poincare (IHP) in Paris. The details about this seminar (such as current schedule, previous and next talks, abstracts, etc.) can be found at this website here.

For the time being, this seminar is an experiment in the sense that IHP allows us to use their rooms from March to June 2014. Of course, if the experiment is a success (i.e., if it manages to gather a non-trivial number of participants interested in flat surfaces and Teichmueller dynamics), then we plan to continue it.

Below the fold, I will reproduce my notes of Jean-Christophe’s talk about a new result together with Stefano Marmi on the cohomological equation for interval exchange transformations of restricted Roth type. Logically, it goes without saying that any errors/mistakes are my entire responsibility.

**1. Introduction **

A classical method to study the properties of (“quasi-periodic”) dynamical system consists into finding an adequate *linearization*, i.e., one seeks a (“smooth”) change of coordinates so that the new dynamical system is “linear”/“algebraic” in some sense (e.g., a rigid rotation on a circle, a translation on a torus, etc.).

Of course, given and a “good candidate” for a linear model of , the problem of finding is *non-linear* (because the *conjugation equation* is non-linear in ). For this reason, it is often the case that before attacking the conjugation equation one studies the following *linear* version

called *cohomological equation* for (where is given and we want to solve for ). In fact, the relationship between the cohomological equation and the conjugation equation was already discussed in this blog (see, e.g., this post), where we emphasized Herman’s Schwartzian derivative trick to convert solutions of the cohomological equation into solutions of the conjugation equation in the context of circle diffeomorphisms.

Today, we will discuss exclusively the existence and regularity of solutions of the cohomological equation for interval exchange transformations (but we will not study the conjugation equation).

In order to motivate the main results in this post, let us recall some of the known theorems about the existence and regularity of solutions of the cohomological equation for rotations on the circle of angle (or, equivalently, an interval exchange transformation of two intervals of lengths and ).

Definition 1We say that an irrational number is ofRoth typewhenever for all there exists such that

for all . Here, means the distance to the closest integer.

Remark 1The nomenclature “Roth type” is motivated by Roth’s theorem stating that any irrational algebraic integer is of Roth type.

Proposition 2 (Russmann, Herman, …)Let be of Roth type. Given and (i.e., is a function on with zero mean), there exists a solution of the cohomological equation

for the rotation of angle on with the property that for all .

In other terms, this result says that we can solve the cohomological equation for circle rotations of Roth type with a loss of ()-derivative for all .

Remark 2The analog of this result in the Sobolev scale (i.e., when and belong to standard Sobolev spaces ) follows from an elementary Fourier analysis (cf. this post). On the other hand, the statement above (in Hölder scale ) requires some extra work, but it is still within the framework of Harmonic Analysis in the sense that one uses Littlewood-Paley decomposition and interpolation inequalities (cf. Herman’s article for more details).

For the sake of comparison, let us give the following statement (where the boldface terms will introduced later):

Theorem 3 (Marmi-Yoccoz)Let be an interval exchange transformation ofrestricted Roth type. Given , there exists , a subspace of codimension and a bounded operator such that

for every .

Of course, since a rotation of the circle is an interval exchange transformation in two intervals, the theorem of Marmi-Yoccoz extends the previous proposition (of Russmann, Herman, …) to a larger important class of quasi-periodic dynamical systems.

Remark 3It is possible to check that a circle rotation of Roth type is an interval exchange transformation of two intervals of restricted Roth type. Furthermore, in this particular context one also can show that .

Remark 4The theorem of Marmi-Yoccoz applies to almost every interval exchange transformation: in fact, the restricted Roth type condition has full measure in the space of interval exchange transformations (with respect to the natural Lebesgue measure obtained by parametrizing interval exchange transformations with fixed combinatorics via the lengths of the permuted intervals).

On the other hand, the loss of derivative in Marmi-Yoccoz theorem is not very good compared to the previous proposition: in fact, the quantity depends on and the definition of restricted Roth type in a *highly non-trivial way*, so that is usually very small and, *a fortiori*, is not close (in general) to the optimal loss in the previous proposition.

Remark 5Here, Jean-Christophe said that it is likely that the definition of restriction Roth type must be changed if one desires an optimal loss of derivatives.

Before explaining the terms in boldface in Marmi-Yoccoz theorem, let us recall some previous related results on cohomological equations for interval exchange transformations and translation flows (the“continuous time analogs” of interval exchange transformations).

First, Forni considered in these two papers here (1997) and here (2007) the cohomological equation for translation flows on translation surfaces (i.e., Forni studied the “continuous time analog” of the cohomological equation for interval exchange transformations). Using several tools from Harmonic Analysis (including *weighted Sobolev spaces*) on *compact surfaces*, Forni managed to construct solutions of the cohomological equation for “almost all” (choice of direction for) translation flows in *any* given translation surface with an optimal loss of derivative (in a weighted Sobolev scale).

Secondly, Marmi, Moussa and Yoccoz considered in these two papers here (2005) and here (2012) showed the existence of *continuous* solutions of the cohomological equation for interval exchange transformations of restricted Roth type. In particular, the new result of Marmi-Yoccoz improves these previous results by asserting that the existence of Hölder continuous () solutions for the cohomological equations studied in their previous papers with Moussa.

As the reader can see, the results of Forni and Marmi-Moussa-Yoccoz have both strong and weak points. On one hand, Forni’s result gives solutions to the cohomological equation for “almost all” translation flows with optimal loss of derivative in Sobolev scale, but the “Diophantine condition” (i.e., the subset of full measure of translation flows) in his theorem is not explicit. On the other hand, the results of Marmi-Moussa-Yoccoz result and Marmi-Yoccoz give solutions to the cohomological equation for interval exchange transformations with a poor gain of regularity in Hölder scale, but their “Diophantine condition” (restricted Roth type) on the interval exchange transformation is “relatively explicit”. Also, it is not easy to compare “directly” these results: even though there is a “natural” notion of restricted Roth type translation flow (in the sense that the return map of the translation flow to an appropriate transverse section is an interval exchange transformation of restricted Roth type) in this paper of Marmi-Moussa-Yoccoz, it is not clear that a restricted Roth type translation flow fits the “Diophantine condition” of Forni.

Remark 6Very roughly speaking, one of the (several) difficulties in relating the Diophantine conditions of Forni and Marmi-Moussa-Yoccoz is related to the application of Oseledets theorem for the Kontsevich-Zorich cocycle: indeed, Oseldets theorem provides a non-explicit set of full measure of points such that the Kontsevich-Zorich cocycle along the Teichmüller flow orbit of these points have a particularly nice behavior (see, e.g., the introduction of this paper of Forni for more explanations). Nevertheless, it is worth to point out that the recent results of Chaika-Eskin give some hope towards relating the Diophantine conditions of Forni and Marmi-Moussa-Yoccoz.

After these comments on Theorem 3 (and some related results), it is time to define the objects involved in the statement of this theorem.

**2. Interval exchange transformations **

Recall that an interval exchange transformation is determined by the following data. Given a finite alphabet with letters, an interval and two partitions into subintervals with

for every , the interval exchange transformation is the piecewise translation sending to . Here, , resp. stands for *top*, resp. *bottom* subintervals, that is, the subintervals of the partition one sees before, resp. after applying . The figure below gives some examples of interval exchange transformations.

We denote by , resp. , the extremities of the subintervals , resp. (), so that and are the extremities of the interval . In particular, , resp. , are the discontinuities of , resp. .

Using these notations, we are ready to introduce the first term marked boldface in Theorem 3:

and is the subspace of zero mean functions in . In concrete terms, is the space of piecewise -functions on that are on the intervals admitting natural extensions to the intervals (but these extensions might disagree at the points ‘s).

Next, we introduce the constants and attached to an interval exchange transformation . An interval exchange transformation can be naturally seen (in many ways) as the first return map of a translation flow on a translation surface (by means of *Masur’s suspension construction* or *Veech’s zippered rectangles construction*): the reader can find more details in this post here (for instance). The translations surfaces obtained from in this way have a genus and a number of conical singularities depending only on . Alternatively, one can define and by combinatorial means (in terms of the cycles of the permutation on induced by the way permutes the subintervals and ).

At this point, the sole undefined term in boldface in the statement of Theorem 3 is “restricted Roth type”. In order to do so, we have to introduce the Rauzy-Veech algorithm and the (discrete version of the) Kontsevich-Zorich cocycle (using this survey of Jean-Christophe as a basic reference).

**3. Rauzy-Veech algorithm, Kontsevich-Zorich cocycle and restricted Roth type **

We say that an interval exchange transformation has a *connection* if there are , such that

Since , resp. , is a discontinuity of , resp. (so that the future orbit of , resp. past orbit of is ill-defined), we see that has a connection whenever it has an orbit that is “blocked” (can not be extended) in the future *and* in the past.

An interval exchange transformation *without* connections are very similar to irrational rotations of the circle: by a result of Keane, any without connections has a minimal dynamics.

Starting with without connections, we denote by the first return map of to the subinterval

It is not hard to check that is also an interval exchange transformation permuting a finite collection , of subintervals of naturally indexed by the alphabet . Furthermore, also has no connections. In the literature, the map is called an elementary step of the Rauzy-Veech algorithm.

Of course, the two facts described in the previous paragraph imply that we can *iterate* this procedure: starting with without connections, by successively applying the elementary steps of the Rauzy-Veech algorithm, one obtains a sequence of interval exchange transformations acting on a decreasing sequence of subintervals . Moreover, it is possible to show that the lengths of the intervals tend to zero as .

For later use, let us observe that, by definition, for any , is the first return map of to .

In terms of the Rauzy-Veech algorithm, the Kontsevich-Zorich cocycle can be described as follows. Given and , we consider the special Birkhoff sum

where is the first return time of (under iterates).

It is possible to check that . In particular, denoting by

we see that

is a linear operator inducing a matrix whose entries have the following dynamical interpretation: is the number of visits of to under -iterates before its return to . The matrices form a linear cocycle (i.e., ) called (discrete) Kontsevich-Zorich cocycle.

The restricted Roth type for an interval exchange transformation is defined in terms of the features of the Kontsevich-Zorich cocycle .

More precisely, we define inductively and is the smallest integer such that for all . It is possible to show that this definition leads to a sequence with as .

We say that has *restricted Roth type* whenever the following four conditions are fulfilled.

- (a)
*Roth type condition*: for each , one hasfor all .

Remark 7The fact that the Roth type condition is satisfied for almost all interval exchange transformations (i.e., for Lebesgue almost all choices of lengths of the intervals ) was checked in this paper of Marmi-Moussa-Yoccoz (see also the paper of Avila-Gouezel-Yoccoz).

Remark 8For sake of comparison, in the case of the rotation of angle on the circle (i.e., interval exchange transformation permuting two intervals of lengths and ), one can check that and where are the entries of the continued fraction expansion of and are the denominators of the best rational approximations of . In particular, the Roth type condition is equivalent to

for all , i.e., is of Roth type.

- (b)
*Spectral gap*: there exists such thatwhere is the subspace of functions with zero mean.

Remark 9The spectral gap property is also satisfied by almost all interval exchange transformations thanks to the work of Veech. In fact, this property is closely related to the non-uniform hyperbolicity of the Teichmueller flow (and the constant is the second Lyapunov exponent of the Kontsevich-Zorich cocycle over the Teichmueller flow).

- (d)
*Hyperbolicity*: the*stable space*of the Kontsevich-Zorich cocycle has dimension .

Remark 10The hyperbolicity property is verified for almost all interval exchange transformations thanks to the work of Forni.

- (c)
*Coherence property*: denoting by the restriction of the Kontsevich-Zorich cocycle to the stable space , and by the action of the Kontsevich-Zorich cocycle on the “center-unstable spaces” , then for each , one hasand

Remark 11The coherence property is also verified for almost all interval exchange transformations: indeed, this is a consequence of Oseledets theorem applied to the Kontsevich-Zorich cocycle (see, e.g., Marmi-Moussa-Yoccoz paper for more details).

Remark 12We called “item (d)” the hyperbolicity property and “item (c)” the coherence property just to keep the same notations of this paper of Marmi-Moussa-Yoccoz (and also because Jean-Christophe did the same during the talk :) ).

At this point, all boldfaced terms in Theorem 3 were defined and now it is time to discuss some points of the proof of this result.

**4. Some steps of the proof of Theorem 3 **

Recall that the quantity is the number of marked points of a translation surface obtained by suspension of the interval exchange transformation . Combinatorially, these marked points can be seen as cycles of a permutation on keeping track of the ‘s one sees when turning counterclockwise around the conical singularities in the translation surface . (See, e.g., this survey of Jean-Christophe for more details)

The permutation allows to define a *boundary operator* ,

where is the set of cycles of , and , resp. , means , resp. .

It is not hard to see that the boundary operator has the following properties:

- ;
- ;
- the restriction of to is the usual boundary operator in relative homology
after appropriate (natural) identifications and .

In this setting, Marmi-Yoccoz deduce Theorem 3 as an immediate consequence of the following more precise statement:

Theorem 4 (Marmi-Yoccoz)Denote by the kernel of the (restriction of the) boundary operator (to ) and consider an arbitrary supplement of in (i.e., ).Let be an interval exchange transformation of restricted Roth type.Then, there are two bounded operators and such that

where and .

Remark 13This theorem says that we can solve the cohomological equation whenever and (i.e., there is no obstruction coming from the boundary operator and the operator ). In the literature, these conditions (or “obstructions”) are called Forni’s distributions.

Closing today’s post, let us give some steps of the proof of Theorem 4.

The first three steps are the following:

- (1) there exists such that for all with (zero mean) one has
- (2) there exists and a bounded operator such that for all one has
where

- (3) by Gottschalk-Hedlund theorem applied to the homeomorphism of the compact space obtained after blowing up the orbits of the discontinuities of and (and by keeping track of what happens once one undo the blowup), one can use the previous steps to write for some
*continuous*function .

In fact, these three steps were performed in this paper of Marmi-Moussa-Yoccoz from 2012 (where they are explained in details).

Remark 14As it turns out, the “non-optimal loss of derivatives” in Marmi-Yoccoz is already present here: in fact, it seems that for an optimal loss of derivatives for the solutions of the cohomological equation in the setting of Marmi-Yoccoz one has to improve the information on the constant appearing in the first step above.

At this stage, it is clear that the proof of Theorem 4 is reduced to show that the function appearing in Step 3 above is Hölder continuous. Here, the main *novelty* introduced by Marmi-Yoccoz is the following fourth step:

- (4) Given an interval , denote by ,
for each , and

Then, there exists such that one has the following “almost recurrence relation”

for the sequence of vectors .

At this point, Jean-Christophe was running out of time so that he decided to skip the proof of this almost recurrence relation (based, of course, on the definition of restricted Roth type) in order to explain how this new step allows to conclude Theorem 4.

First, one uses the fact that is continuous to check that (this is not difficult since we are not asking for moduli of continuity/rate of convergence).

By combining this information with Step 4 above, they can show that there exists such that

From this inequality, it is not difficult to deduce a Hölder modulus of continuity for by “interpolation” of the information at the extremities of .

Finally, the proof of the estimate (1) itself consists into three steps.

One constructs first a vector where is the intersection of with the kernel of and is a natural *Kontsevich-Zorich cocycle invariant* symplectic form on (related to the intersection form on ).

After this, Marmi-Yoccoz introduce the vector .

Then, Marmi-Yoccoz use the coherence property (c) in the definition of restricted Roth type (among several other facts) to show the analog of (1) for . Moreover, they can check that this estimate on can be transferred to . Furthermore, by exploiting the symplecticity of the Kontsevich-Zorich cocycle (among several other facts), Marmi-Yoccoz show the analog of (1) for the vectors imply the estimate (1) for the vectors , so that the argument is complete.

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Given the very interesting program of this conference, it was not surprising that Amphithéâtre Hermite (where the talks were delivered) was always full.

Today, we will discuss one of the talks of this conference, namely, the talk “On the continuity of Lyapunov spectrum for random products” of Alex Eskin about his joint work (in preparation) with Artur Avila and Marcelo Viana.

As usual, all mistakes/errors in this post are entirely my responsibility.

Remark 1A video of a talk of Artur Avila on the same subject can be found here.

Update [February 11, 2014]:Last Friday, I was lucky enough to get some extra explanationsconcerning “costs of couplings” directly from Alex. At the end of this post (see the “Epilogue”), I will try to briefly summarize what I could understand from this conversation.

**1. Introduction **

Let be a probability measure on , e.g., where is a (non-trivial) probability vector (i.e., and for all ) and are Dirac masses at .

Consider the random walk on induced by , i.e., let , , and, for each , , put

Remark 2Of course, the intuition here is that the samples , , are describing a random walk on whenever we perform a random choice of with respect to (or, equivalently, random choices of ‘s with probability distribution ).

In this context, the Oseledets multiplicative ergodic theorem says that:

Theorem 1 (Oseledets)For -almost every , one has

where is a symmetric matrix with eigenvalues . (Here, is the transpose matrix of , and denotes the non-negative symmetric matrix such that .)

The numbers are called Lyapunov exponents.

Geometrically, Oseledets theorem says that the random walk almost surely tracks a geodesic of speed of the symmetric space (where is a maximal compact subgroup of ).

Remark 3The top Lyapunov exponent can be recovered by the formula

for -a.e. , and the remaining Lyapunov exponents can be recovered by the following standard trick/observation: the top Lyapunov exponent of the action of on the -th exterior power is . For this reason, it is often (but not always!) the case that the results about the top Lyapunov exponent also provide information about all Lyapunov exponents.

Historically, the first results about the Lyapunov exponents of random products concerned their multiplicities for a *fixed* probability distribution . A prototypical theorem in this direction is the following result of Guivarch-Raugi and Goldsheid-Margulis providing sufficient conditions for the simplicity (multiplicity ) of Lyapunov exponents.

Definition 2We say that isnotstrongly irreducible whenever there exists a finite collection of subspaces of such that

for all .

Definition 3We say that is proximal if there exists such that has distinct eigenvalues. (Here, is the Zariski closure of the group generated by .)

Remark 4If is Zariski dense in , then is strongly irreducible and proximal.

Theorem 4 (Guivarch-Raugi, Goldstein-Margulis)

- 1) If is strongly irreducible and proximal, then (i.e., the top Lyapunov exponent is simple/has multiplicity );
- 2) If is Zariski dense in , then .

**2. Statement of the main result **

In their work, Avila, Eskin and Viana consider how the Lyapunov exponents change when the probability distribution *varies*. Among the results that they will prove in their forthcoming article is:

Theorem 5 (Avila-Eskin-Viana)Suppose is afixedprobability vector, and consider the probability measures

whose supportvaries. Then, for each , the Lyapunov exponent is a continuous function of .

Remark 5This statement looks innocent, but it is known that Lyapunov exponents do not vary continuously (only upper semi-continuously) “in general”. See, e.g., this article of Bochi (and the references therein) for more details.

**3. Previous works and related results **

The theorem of Avila-Eskin-Viana generalizes to any dimension the work of Bocker-Neto and Viana in dimension :

Theorem 6 (Bocker-Neto-Viana)For a fixed probability vector , the two Lyapunov exponents of

depend continuously on .

On the other hand, if one decides to fix the support and to vary the vector of probabilities, then Peres showed in 1991 that:

Theorem 7 (Peres)Let us fix the support . Then, the simple Lyapunov exponents of

are locally real-analytic function of . More precisely, given and a probability vector such that the th Lyapunov exponent is simple (i.e., multiplicity ), then the th Lyapunov exponent is a real-analytic function of near .

The formula for -a.e. for the top Lyapunov exponent is not very useful to study how Lyapunov exponents vary with because the notion of “-a.e. ” changes radically with .

A slightly more useful formula was found by Furstenberg:

where and is a -stationary measure (i.e., is invariant in average with respect to , that is, ) on the projective space of lines in .

Of course, the cocycle depends nicely on and , but the dependence on of the stationary measure in Furstenberg’s formula is not obvious to determine. In particular, one needs to “feed” Furstenberg’s formula with extra information in order to deduce continuity of the top Lyapunov exponent in a given setting.

For example, if one feeds the following remark

Remark 6If is strongly irreducible and proximal, then the stationary measure on is unique.

to Furstenberg’s formula, then one can deduce:

Proposition 8Suppose (in the weak-* topology), is proximal and strongly irreducible. Then, .

*Proof:* Denote by the sequence of stationary measures associated to in Furstenberg’s formula. It is not hard to check that any accumulation of the sequence is -stationary. By the previous remark, has an unique stationary measure on , so that any accumulation of coincides with the stationary measure in Furstenberg’s formula for . In other words, , and the desired proposition now follows immediately from Furstenberg’s formula.

Remark 7Le Page showed that the conclusion of the previous proposition can be improved from continuity to real-analyticity. However, in general (without strong irreducibility and proximality of ), one can not expect anything better than Hölder continuity.

**4. Some ideas of the proof of Avila-Eskin-Viana theorem **

Let us simplify the exposition by considering the following toy case: we are given two sequences of matrices

and

and we want to show that the top Lyapunov exponents of the probabilities

converge to

The projective actions of the matrices and on the projective circle are of “north pole–south pole type”: there are two fixed points and corresponding to the directions of the coordinate axes and of and the points of are either attracted or repelled towards and under the actions of and . In particular, one can infer from this that an arbitrary -stationary measure on has the form

with .

Therefore, if we denote by the -stationary measures coming from Furstenberg’s formula, then

and our goal is to show that . However, there is not so easy as it seems (in the sense that naive methods don’t work well) and one has to look for appropriate tools.

In this direction, the notion of *Margulis function* comes at hand. Given a probability measure on a group acting on a space , let

be the Markov operator associated to . We say that is a Margulis function if:

- 1)
- 2) on a “negligible set”
- 3) there are constants and such that , i.e., when is large at a point (a step of a -random walk approaches ), the value of at the -images of this point decrease in average (the next step of a -random walk tend to get far from ).

Coming back to toy case, it is possible to show that for the function given by

is a Margulis function for .

This type of information is useful to show simplicity of the Lyapunov exponents of , but it does not help us to show the continuity statement or . In fact, the difficulty comes from the fact that is not a Margulis function of because the south pole of is changing location (even though they are close to ), so that a *single* Margulis function is not capable of assigning the value to all of the south poles of without being trivial.

Here, one can try to overcome the technical obstacle of the moving south poles of by considering the diagonal action of on and by introducing the function

for and close to . As it turns out, this function is a good candidate of Margulis function for in the sense that the inequality in item 3) involving the Markov operator is satisfied *near* , and it seems that we are doing some progress.

Unfortunately, we made no progress at all with the idea in the previous paragraph: indeed, the technology of Margulis functions requires *globally* defined functions and so far we were able only to exhibit *locally* defined functions (in a neighborhood of ).

At this point, the basic idea of Avila-Eskin-Viana is the introduction of *measure-theoretical* analogs of Margulis functions. In other terms, they want to replace “functions” by “measures” to get objects that are slightly more flexible but still capable of doing the same job than Margulis functions.

The measure-theoretical analog of Margulis functions are called *couplings* with finite *costs*. Concretely, we say that a probability measure on is a coupling of to itself if the projection of to both factors is . Given a coupling of to itself, we define its cost as:

where is an adequate small neighborhood of .

In this setting, we can see that the task of showing is reduced to find a large constant and a sequence of couplings of to itself such that

for all . Indeed, this is so because the cost of coupling to itself is and thus has finite cost only when .

At this point, the time of Alex Eskin was essentially out and he concluded by saying that the main point is that finding couplings with finite costs is *easier* than building globally defined Margulis functions, and the desired couplings with uniformly bounded costs could be found by analyzing the analog of item 3) in the definition of Margulis functions for couplings of to itself with *optimal* (minimal possible) costs.

**5. Epilogue**

Let us try to give more explanations to the discussion in the previous 6 paragraphs above (following my conversation with Alex Eskin [or what I can remember of it...]).

We start by selecting the small neighborhood of so that the limit stationary measure gives mass

to .

Then, we restrict our measures to and we *change *the dynamics so that these restrictions are stationary: formally, we replace the Markov operator by an adequate “local transfer operator” such that is -stationary.

In these terms, the “local version”

of the “usual candidate to Margulis function” seems to be a Margulis function at first sight, but unfortunately it does not satisfies item 3). Indeed, the *pointwise* estimates of the form

with and do not hold *always* because there are *some* couples of points that are pushed *together* towards despite the fact that the *probability* of this event is *small*.

For this reason, Avila-Eskin-Viana replace “functions” by “measures” with the idea that this probabilistic tendency felt by most couples of getting away from is better expressed as estimates for measures than pointwise estimates for functions.

More concretely, by selecting an appropriate subinterval , one can see that the -measure of the set

of elements of pushing a point towards is . From this information, it is not difficult to construct some measures on such that projects to on both factors and . From the measures , one obtains some couplings with finite costs.

However, this is not quite the end of the history: we need couplings whose costs are *uniformly bounded* for all . Here, the trick is to study couplings with *optimal* costs (i.e. with smallest possible costs). In fact, by applying the “dynamics” to , one has the following analogue of item 3) in the definition of Margulis functions:

for some universal constants and (thanks to the probabilistic tendency of most couples of points to get pushed away from ). On the other hand, since has optimal (smallest) cost, we conclude that

that is,

In other terms, the analog for measures of item 3) in the definition of Margulis functions allows to check that the costs of the sequence optimal cost couplings are uniformly bounded by , as desired.

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Theorem 1 (Burns-Masur-Wilkinson)Suppose that:Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously. Denote by the metric completion of and the boundary of .

- (I) the universal cover of is
geodesically convex, i.e., for every , there exists an unique geodesic segmentinconnecting and .- (II) the metric completion of is
compact.- (III) the boundary is
volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfiesfor every .

- (IV) has
polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial boundfor every .

- (V) has
polynomially controlled injectivity radius, i.e., there are constants and such thatfor every (where denotes the injectivity radius at ).

- (VI) The
first derivative of the geodesic flowispolynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :Then, the Liouville (volume) measure of is finite, the geodesic flow on the unit cotangent bundle of is defined at -almost every point for all time , and the geodesic flow is

non-uniformly hyperbolic(in the sense of Pesin’s theory) andergodic.

Actually, the geodesic flow is Bernoulli and, furthermore, its metric entropy is positive, finite and is given by Pesin’s entropy formula (i.e., it is equal to the sum of positive Lyapunov exponents of counted with multiplicities).

However, since the second post of this series was dedicated to the discussion of items (I), (II) and (III) above for the Weil-Petersson (WP) metric, we think it is natural that this third post provides a discussion of items (IV), (V) and (VI) for the Weil-Petersson metric (thus completing the proof of Burns-Masur-Wilkinson theorem of ergodicity of the Weil-Petersson geodesic flow modulo the proof of their ergodicity criterion).

For this reason, we will continue the discussion of the geometry of the Weil-Petersson metric in this post while leaving the proof of Burns-Masur-Wilkinson ergodicity criterion for the next two posts of this series.

The organization of today’s post is very simple: it is divided in three sections where the items (IV), (V) and (VI) for the Weil-Petersson metric are discussed.

**1. The curvatures of the Weil-Petersson metric **

The item (IV) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) asks for polynomial bounds in the sectional curvatures and their first two derivatives.

In the context of the Weil-Petersson (WP) metric, the desired polynomial bounds on the sectional curvatures follow from the work of Wolpert.

** 1.1. Wolpert’s formulas for the curvatures of the WP metric **

This subsection gives a *compte rendu* of some estimates of Wolpert for the behavior of the WP metric near the boundary of the Teichmüller space .

Before stating Wolpert’s formulas, we need an *adapted* system of coordinates (called *combined length basis* in the literature) near the strata , , of , where is the curve complex of (introduced in the previous post).

Denote by the set of pairs (“basis”) where is a simplex of the curve complex and is a collection of simple closed curves such that each is disjoint from all . Here, we allow that two curves *intersect* (i.e., one might have ) and also the case is *not* excluded.

Following the nomenclature introduced by Wolpert, we say that is a *combined length basis* at a point whenever the set of tangent vectors

is a basis of , where is the length parameter in the Fenchel-Nielsen coordinates and .

Remark 1The length parameters and their square-roots are natural for the study of the WP metric: for instance, Wolpert showed that these functions are convex along WP geodesics (see, e.g., these papers of Wolpert and this paper of Wolf).

The name *combined length basis* comes from the fact that we think of as a combination of a collection of *short* curves (indicating the boundary stratum that one is close to), and a collection of *relative* curves to allowing to complete the set into a basis of the tangent space to in which one can write nice formulas for the WP metric.

This notion can be “extended” to a stratum of as follows. We say is a *relative basis* at a point whenever and the length parameters is a *local* system of coordinates for near .

Remark 2The stratum is (isomorphic to) a product of the Teichmüller spaces of the pieces of . In particular, carries a “WP metric”, namely, the product of the WP metrics on the Teichmüller spaces of the pieces of . In this setting, is a relative basis at if and only if is a basis of .

Remark 3Contrary to the Fenchel-Nielsen coordinates, the length parameters associated to a relative basis might not be aglobalsystem of coordinates for . Indeed, this is so because we allow the curves in to intersect non-trivially: geometrically, this means that there are points in where the geodesic representatives of such curves meet orthogonally, and, at such points , the system of coordinates induced by meet a singularity.

The relevance of the concept of combined length basis to the study of the WP metric is explained by the following theorem of Wolpert:

Theorem 2 (Wolpert)For any point , , there exists a relative length basis . Furthermore, the WP metric can be written as

where the implied comparison constant is uniform in a neighborhood of .In particular, there exists a neighborhood of such that is a combined length basis at any .

The statement above is just the beginning of a series of formulas of Wolpert for the WP metric and its sectional curvatures written in terms of the local system of coordinates induced by a combined length basis .

In order to write down the next list of formulas of Wolpert, we need the following notations. Given an arbitrary collection of simple closed curves on , we define

where . Also, given a constant and a basis , we will consider the following (Bers) region of Teichmüller space:

Wolpert provides several estimates for the WP metric and its sectional curvatures in terms of the basis , and , , which are uniform on the regions .

Theorem 3 (Wolpert)Fix . Then, for any , and any and , the following estimates hold uniformly on

- where is the Kronecker delta.
- and, furthermore, extends continuosly to the boundary stratum .
- the distance from to the boundary stratum is
- for any vector ,
- and
- extends continuously to the boundary stratum
- the sectional curvature of the complex line (real two-plane) is
- for any quadruple , distinct from a curvature-preserving permutation of , one has
and, moreover, each of the form or introduces a multiplicative factor in the estimate above.

These estimates of Wolpert gives a very good understanding of the geometry of the WP metric in terms of combined length basis. For instance, one infers from the last two items above that the sectional curvatures of the WP metric along the complex lines converge to with speed as one approaches the boundary stratum , while the sectional curvatures of the WP metric associated to quadruples of the form with and converge to with speed *at least*.

In particular, these formulas of Wolpert allow to show “1/3 of item (IV)” for the WP metric, that is,

for all .

Remark 4Observe that the formulas of Wolpert provideasymmetricinformation on the sectional curvatures of the WP metric: indeed, while we have precise estimates on how these sectional curvarutures can approach , the same is not true for the sectional curvatures approaching zero (where one disposes of lower bounds but no upper bounds for the speed of convergence).

Remark 5From the discussion above, we see that there are sectional curvatures of the WP metric on approaching zero whenever contains two distinct curves. In other words, the WP metric has sectional curvatures approaching zero whenever the genus and the number of punctures of satisfy , i.e., except in the cases of once-punctured torii and four-times puncture spheres . This qualitative difference on the geometry of the WP metric on in the cases and (i.e., or ) will be important in the last post of this series when we will discuss the rates of mixing of the WP geodesic flow.

Remark 6As Wolpert points out in this paper here, these estimates permit to think of the WP metric on the moduli space in a -neighborhood of the cusp at infinity as a -pertubation of the metric of the surface of revolution of the profile modulo multiplicative factors of the form .

Now, we will investigate the remaining “2/3 of item (IV)” for the WP metric, i.e., polynomial bounds for the first two derivatives and of the curvature operator of the WP metric.

** 1.2. Bounds for the first two derivatives of WP metric **

As it was *recently* pointed out to us by Wolpert (in a private communication), it is possible to deduce very good bounds for the derivatives of the WP metric (and its curvature tensor) by refining the formulas for the WP metric in some of his works.

Nevertheless, by the time the article of Burns, Masur and Wilkinson was written, it was not clear at all that the delicate calculations of Wolpert for the WP metric could be extended to provide useful information about the derivatives of this metric.

For this reason, Burns, Masur and Wilkinson decided to implement the following alternative strategy.

At first sight, our task reminds the setting of Cauchy’s inequality in Complex Analysis where one estimates the derivatives of a holomorphic function in terms of given bounds for the -norm of this function via the Cauchy integral formula. In fact, our current goal is to estimate the first two derivatives of a “function” (actually, the curvature tensor of the WP metric) defined on the complex-analytic manifold knowing that this “function” already has nice bounds (cf. the previous subsection).

However, one can *not* apply the argument described in the previous paragraph *directly* to the curvature tensor of the WP metric because this metric is *only* a real-analytic (but *not* a complex-analytic/holomorphic) object on the complex-analytic manifold .

Fortunately, Burns, Masur and Wilkinson observed that this idea of using the Cauchy inequalities could still work *after* one adds some results of McMullen into the picture. In a nutshell, McMullen showed that the WP metric is closely related to a *holomorphic* object: very roughly speaking, using the so-called Bers simultaneous uniformization theorem, one can think of the Teichmüller space as a *totally real* submanifold of the so-called quasi-Fuchsian locus , and, in this setting, the Weil-Petersson symplectic -form is the restriction to of the differential of a *holomorphic* -form globally defined on the quasi-Fuchsian locus . In particular, it is possible to use Cauchy’s inequalities to the holomorphic object to get some estimates for the first two derivatives of the WP metric.

Remark 7Acaricatureof the previous paragraph is the following. We want to estimate the first two derivatives of a real-analytic function (“WP metric”) knowing some bounds for the values of . In principle, we can not do this by simply applying Cauchy’s estimates to , but in our context we know (“by the results of McMullen”) that the natural embedding of as a totally real submanifold of allows to think of as the restriction of a holomorphic function and, thus, we can apply Cauchy inequalities to to get some estimates for .

In what follows, we will explain the “Cauchy inequality” idea of Burns, Masur and Wilkinson in two steps. Firstly, we will describe the embedding of into the quasi-Fuchsian locus and the holomorphic -form of McMullen whose differential restricts to the WP symplectic -form on . After that, we will show how the Cauchy inequalities can be used to give the remaining “2/3 of item (IV)” for the WP metric.

**1.2.1. Quasi-Fuchsian locus and McMullen’s -forms **

Given a hyperbolic Riemann surface , , the *quasi-Fuchsian locus* is defined as

where is the *conjugate* Riemann surface of , i.e., is the quotient of the *lower-half plane* by . The *Fuchsian locus* is the image of under the *anti-diagonal* embedding

Geometrically, we can think of elements as follows. Recall that and are related to and via (extremal) quasiconformal mappings determined by the solutions of Beltrami equations associated to -invariant Beltrami differentials (coefficients) and on and . Now, we observe that and live naturally on the Riemann sphere . Since the real axis/circle at infinity/equator has zero Lebesgue measure, we see that and induce a Beltrami differential on . By solving the corresponding Beltrami equation, we obtain a quasiconformal map on and, by conjugating, we obtain a quasi-Fuchsian subgroup

i.e., a Kleinian subgroup whose domain of discontinuity consists of two connected components and such that and .

The following picture summarizes the discussion of the previous paragraph:

Remark 8The Jordan curve given by the image of the equator under the quasiconformal map is “wild” in general, e.g., it has Hausdorff dimension (as the picture above tries to represent). In fact, this happens because a typical quasiconformal map is merely a Hölder continuous, and, hence, it might send “nice” curves (such as the equator) into curves with “intricate geometries” (see, e.g., the three external links of the Wikipedia article on quasi-Fuchsian groups).

The data of the quasi-Fuchsian subgroup attached to permits to assign (marked) *projective structures* to and . More precisely, by writing and with and , we are equipping and with projective structures, that is, atlases of charts to whose changes of coordinates are Möebius transformations (i.e., elements of ). Furthermore, by recalling that and come with markings and (because they are points in Teichmüller spaces), we see that the projective structures above are marked.

In summary, we have a natural *quasi-Fuchsian uniformization* map

assigning to the marked projective structures

Here, is the “Teichmüller space of projective structures” on , i.e., the space of “Teichmüller” equivalence classes of marked projective structures where two marked projective structures and are “Teichmüller” equivalent whenever there is a projective isomorphism homotopic to .

Remark 9The procedure (due to Bers) of attaching a quasi-Fuchsian subgroup to a pair of hyperbolic surfaces and is called Bers simultaneous uniformization because the knowledge of allows to equipat the same timeand with natural projective structures.

Note that is a section of the natural projection

obtained by sending each pair of (marked) projective structures , , , to the unique pair of (marked) compatible conformal structures , , .

We will now describe how the (affine) structure of the fibers of the projection and the section can be used to construct McMullen’s primitives/potentials of the Weil-Petersson symplectic form .

Given two projective structures in the same of the projection , one can measure how far apart from each other are and using the so-called Schwarzian derivative.

More precisely, the fact that and induce the same conformal structure means that the charts of atlases associated to them can be thought as some families of maps and from (small) open subsets to the Riemann sphere , and we can measure the “difference” by computing how “far” from a Möebius transformation (element of ) is .

Here, given a point , one observes that there exists an *unique* Möebius transformation such that and *coincide* at up to *second order* (i.e., and have the same value and the same first and second derivatives at ). Hence, it is natural to measure how far from a Möebius transformation is by understanding the difference between the *third derivatives* of and at , i.e., .

Actually, this is *almost* the definition of the *Schwarzian derivative*: since the derivatives of and map to , in order to recover an object from to *itself*, it is a better idea to “correct” with , i.e., we define the Schwarzian derivative of and at as

Here, the factor shows up for historical reasons (that is, this factor makes coincide with the classical definition of Schwarzian derivative in the literature).

By definition, the Schwarzian derivative is a field of quadratic forms on (since its definition involves taking third order derivatives). In other terms, is a *quadratic differential* on , that is, the “difference” between two projective structures in the same fiber of the projection is given by a quadratic differential . In particular, the fibers are affine spaces modeled by the space of quadratic differentials on .

Remark 10The reader will find more explanations about the Schwarzian derivative in Section 6.3 of Hubbard’s book.

Remark 11The idea of “measuring” the distance between projective structures (inducing the same conformal structure) by computing how far they are from Möebius transformations via the Schwarzian derivative is close in some sense to the idea of measuring the distance between two points in Teichmüller space by computing the eccentricities of quasiconformal maps between these points.

Using this affine structure on and the fact that is the cotangent space of at , we see that, for each , the map

defines a (holomorphic) -form on . Note that, by letting vary and by fixing , we have a map given by

Since (so that ) and , we can think of as a (holomorphic) -form on .

For later use, let us notice that the -form is *bounded* with respect to the Teichmüller metric on . Indeed, this is a consequence of *Nehari’s bound* stating that if is a round disc (i.e., the image of the unit disc under a Möebius transformation) equipped with its hyperbolic metric and is an injective complex-analytic map, then

In this setting, McMullen constructed primitives/potentials for the WP symplectic form as follows. The Teichmüller space sits in the quasi-Fuchsian locus as the *Fuchsian locus* where is the *anti-diagonal* embedding

By pulling back the -form under , we obtain a bounded -form

Remark 12This form is closely related to a classical object in Teichmüller theory calledBers embedding: in our notation, the Bers embedding is

McMullen showed that the bounded -forms are primitives/potentials of the WP symplectic -form , i.e.,

See also Section 7.7 of Hubbard’s book for a nice exposition of this theorem of McMullen. Equivalently, the restriction of the holomorphic -form to the Fuchsian locus (a *totally real* sublocus of ) permits to construct (Teichmüller bounded) primitives for the WP symplectic form on .

At this point, we are ready to implement the “Cauchy estimate” idea of Burns-Masur-Wilkinson to deduce bounds for the first two derivatives of the curvature operator of the WP metric.

**1.2.2. “Cauchy estimate” of after Burns-Masur-Wilkinson**

Following Burns-Masur-Wilkinson, we will need the following local coordinates in :

Proposition 4There exists an universal constant such that, for any , one has a holomorphic embedding

of the Euclidean unit polydisc (where ) sending to and satisfying

where is the Teichmüller norm and is the Euclidean norm on .

This result is proven in this paper of McMullen here.

Also, since the statement of Proposition 4 involves the Teichmüller norm and we are interested in the Weil-Petersson norm , the following comparison (from Lemma 5.4 of Burns-Masur-Wilkinson paper) between and will be helpful:

Lemma 5There exists an universal constant such that, for any and any cotangent vector , one has

where is the systole of (i.e., the length of the shortest closed simple hyperbolic geodesics on ). In particular, for any and any tangent vector , one has

*Proof:* Given , let us write with is “normalized” to contain the element where .

Fix a Dirichlet fundamental domain of the action of centered at the point .

By the *collaring theorem* stating that a closed simple hyperbolic geodesic of length has a collar [tubular neighborhood] of radius isometrically embedded in and two of these collars and are disjoint whenever and are disjoint (see, e.g., Theorem 3.8.3 in Hubbard’s book), we have that the union of isometric copies of contains a ball of fixed (universal) radius around any point .

By combining the Cauchy integral formula with the fact stated in the previous paragraph, we see that

Since the hyperbolic metric is bounded away from on , we can use the -norm estimate on above to deduce that

for some constant . This completes the proof of the lemma.

Remark 13The factor in the previous lemma can be replaced by via a refinement of the argument above. However, we will not prove this here because this refined estimate is not needed for the proof of the main results of Burns-Masur-Wilkinson.

Using the local coordinates from Proposition 4 (and the comparison between Teichmüller and Weil-Petersson norms in the previous lemma), we are ready to use Cauchy’s inequalities to estimate “‘s” of the WP metric. More concretely, denoting by “centered at some ” in Proposition 4, let , and consider the vector fields

on . In setting, we denote by the “‘s” of the WP metric in the local coordinate and by the inverse of the matrix .

Proposition 6There exists an universal constant such that, for any , the pullback of the WP metric local coordinate “centered at ” in Proposition 4 verifies the following estimates:

and

for all , and , .

*Proof:* The first inequality

follows from Proposition 4 and Lemma 5. Indeed, by letting , we see from Proposition 4 and Lemma 5 that

Since

we deduce that

i.e., .

For the proof of second inequality (estimates of the -derivatives of ‘s), we begin by “rephrasing” the construction of McMullen’s -form in terms of the local coordinate introduced in Proposition 4.

The composition of the local coordinate with the anti-diagonal embedding of the Teichmüller space in the quasi-Fuchsian locus can be rewritten as

where is the anti-diagonal embedding

and the local coordinate given by

In this setting, the pullback by of the holomorphic -form gives a holomorphic -form on . Moreover, since the Euclidean metric on is comparable to the pullback by of the Teichmüller metric (cf. Proposition 4), is bounded in Teichmüller metric and where , we see that

where and is a holomorphic bounded (in the Euclidean norm) -form on .

Let us write in complex coordinates , where are bounded holomorphic functions. Hence,

and, a fortiori,

Since is the Kähler form of the metric , we see that the coefficients of are linear combinations of the -pullbacks of and . Because are (universally) bounded holomorphic functions, we can use Cauchy’s inequalities to see that the derivatives of are (universally) bounded at any with . It follows from the boundedness of the (*non-holomorphic*) anti-diagonal embedding that the -derivatives of ‘s satisfy the desired bound.

The estimates in Proposition 6 (controlling the WP metric in the local coordinates constructed in Proposition 4) permit to deduce the remaining “2/3 of item (IV)” for the WP metric:

Theorem 7 (Burns-Masur-Wilkinson)There are constants and such that, for any , the curvature tensor of the WP metric satisfies

*Proof:* Fix and consider the local coordinate provided by Proposition 4. Since and are uniformly bounded, our task is reduced to estimate the first two derivatives of the curvature tensor of the metric at the origin .

Recall that the Christoffel symbols of are

or

in Einstein summation convention, and, in terms of the Christoffel symbols, the coefficients of the curvature tensor are

Therefore, we see that the coefficients of the -derivative is a polynomial function of and the first partial derivatives whose “degree” in the “variables” is (because of the formula ).

By Proposition 6, each has order and the first partial derivatives of at are bounded by a constant depending only on . It follows that

and, consequently,

This completes the proof.

At this point, we have that Theorems 3 and 7 imply the validity of item (IV) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) for the WP metric.

Remark 14In a very recent private communication, Wolpert indicated that it is possible to derive theThe estimates for the derivatives of the curvature tensor appearing in the proof of Theorem 7 arenotsharp with respect to the exponent . For instance, the WP metric on the moduli space of once-punctured torii has curvature where is the WP distance between and the boundary , so that one expects tha the -derivatives of the curvature behave like (i.e., the exponent above should be ).sharpestimates of the form

for the derivatives of the curvature tensor of the WP metric from his works.

**2. Injectivity radius of the Weil-Petersson metric **

In this short section, we will verify item (V) of Burns-Masur-Wilkinson ergodicity criterion (Theorem 1) for the WP metric, i.e.,

Theorem 8There exists a constant such that for all , , one has the following polynomial lower bound on the injectivity radius of the WP metric at :

The proof of this result also relies on the work of Wolpert. More precisely, Wolpert showed in this paper here that there exists a constant such that, for any and with ,

where is the Abelian subgroup of the “level ” mapping class group generated by the Dehn twists about the curves .

This reduces the proof of Theorem 8 to the following lemma:

Lemma 9There exists an universal constant with the following property. For each , there exists such that, for any with

for some non-trivial , one can find so that and for some .

*Proof:* We begin the proof of the lemma by recalling that the mapping class group acts on in a properly discontinuous way with no fixed points. Therefore, for each , there exists such that if for some non-trivial (i.e., some non-trivial element of the mapping class group has an “almost fixed point”), then (i.e., the “almost fixed point” is close to the boundary of ).

Let us show now that in the setting of the previous paragraph, for some .

In this direction, let be the product of and the maximal orders of all finite order elements of the mapping class groups of “lower complexity” surfaces. By contradiction, let us assume that there exist infinite sequences , , , such that for some and

but for all , .

Passing to a subsequence (and applying appropriate elements of ), we can assume that the sequence converges to some noded Riemann surface . Because as , we see that ,for each ,

It follows that, for all sufficiently large, sends any curve to another curve . Therefore, for each sufficiently large, there exists

such that *fixes* each (i.e., is a *reducible* element of the mapping class group). By the Nielsen-Thruston classification of elements of the mapping class groups, the restrictions of to each piece of are given by compositions of Dehn twists about the boundary curves with either a pseudo-Anosov or a periodic (finite order) element (in a surface of “lower complexity” than ).

It follows that we have only two possibilities for : either the restriction of to *all* pieces of are compositions of Dehn twists about certain curves in and *finite order* elements, or the restriction of to *some* piece of is the composition of Dehn twists about certain curves in and a *pseudo-Anosov* element.

In the first scenario, by the definition of , we can replace by an adequate power with to “kill” the finite order elements and “keep” the Dehn twists. In other terms, (with ), a contradiction with our choice of the sequence .

This leaves us with the second scenario. In this case, by definition of , we can replace by an adequate power with such that the restriction of to some piece of is pseudo-Anosov. However, Daskalopoulos and Wentworth showed that there exists an *uniform* positive lower bound for

when is pseudo-Anosov on some piece of . Since and is an universal constant, it follows that there exists an uniform positive lower bound for

for all sufficiently large, a contradiction with our choice of the sequences and .

These contradictions show that the sequences and with the properties described above can’t exist.

This completes the proof of the lemma.

**3. First derivative of the Weil-Petersson flow **

This section concerns the verification of item (VI) of Theorem 1 for the WP flow . More precisely, we will show the following result:

Theorem 10There are constants , , and such that

for any and any with

The proof of this result in Burns-Masur-Wilkinson paper is naturally divided into two steps.

In the first step, one shows the following *general* result providing an estimate for the first derivative of the geodesic flow on *arbitrary negatively curved* manifold:

Theorem 11Then,Let be a negatively curved manifold. Consider a geodesic where and suppose that for every the sectional curvatures of any plane containing is greater than for some Lipschitz function .

where is the solution of Riccati equation

with initial data .

We postpone the proof of this theorem to the next post when we will introduce Sasaki metric, Jacobi fields and matrix Riccati equation (among other classical objects) in our way of showing the “abstract” Burns-Masur-Wilkinson ergodicity criterion for geodesic flows.

In the second step, one uses the works of Wolpert to exhibit an adequate bound for the sectional curvatures of the WP metric along WP geodesics . More concretely, one has the following theorem:

Theorem 12There are constants and such that for any and any geodesic segment there exists a positive Lipschitz function with

- (a) for all ;
- (b) is -controlled in the sense that has a right-derivative satisfying
- (c) ;
- (d) .

Here, denotes the distance between the geodesic segment and .

Using Theorems 11 and 12, we can easily complete the proof of Theorem 10 (i.e., the verification of item (VI) of Burns-Masur-Wilkinson ergodicity criterion for the WP metric):

*Proof:* Denote by the “WP curvature bound” function provided by Theorem 12 and let be the solution of Riccati’s equation

with initial data .

Since is -controlled (in the sense of item (b) of Theorem 12), it follows that for all : indeed, this is so because , and, if for some , then

Therefore, by applying Theorem 11 in this setting, we deduce that

for and some constant . This completes the proof of Theorem 10.

Closing this post, let us *sketch* the proof of Theorem 12 (while referring to Subsection 4.4 of Burns-Masur-Wilkinson paper [especially Proposition 4.22 of this article] for more details).

We start by describing how the function is defined. For this sake, we will use Wolpert’s formulas in Theorem 3 above.

More precisely, since the sectional curvatures of the WP metric approach or only near the boundary, we can assume that our geodesic segment in the statement of Theorem 12 is “relatively close” to a boundary stratum , (formally, as Burns-Masur-Wilkinson explain in page 883 of their paper, one must use Proposition 4.7 of their article to produce a nice “thick-thin” decomposition of the Teichmüller space ).

In this setting, for each , we consider the functions and

(where ) along our geodesic segment , . Notice that it is natural to consider these functions in view of the statements in Wolpert’s formulas in Theorem 3.

The WP sectional curvatures of planes containing the tangent vectors to are controlled in terms of and . Indeed, given , we can use a combined length basis to write

Similarly, let us write

By Theorem 3, we obtain the following facts. Firstly, since and are WP-unit vectors, the coefficients are

Secondly, by definition of , we have that

Finally,

In summary, Wolpert’s formulas (Theorem 3) imply that

(cf. Lemma 4.17 in Burns-Masur-Wilkinson paper).

Now, we want convert the expressions into a positive Lipschitz function satisfying the properties described in items (b), (c), and (d) of Theorem 12, i.e., a -controlled function with appropriately bounded total integral and values at and . We will not give full details on this (and we refer the curious reader to Subsection 4.4 of Burns-Masur-Wilkinson paper), but, as it turns out, the function

where is the (unique) time with for all and is a sufficiently large constant satisfies the conditions in items (a), (b), (c) and (d) of Theorem 12. Here, the basic idea is these properties are consequences of the features of two ODE’s (cf. Lemmas 4.15 and 4.16 in Burns-Masur-Wilkinson paper) for and . For instance, the verification of item (a) (i.e., the fact that controls certain WP sectional curvatures along ) relies on the fact that these two ODE’s permit to prove that

for some sufficiently large constant . In particular, by plugging this into (1), we obtain that

i.e.,, the estimate required by item (a) of Theorem 12.

Concluding this sketch of proof of Theorem 12, let us indicate the two ODE’s on and .

Lemma 13 (Lemma 4.15 of Burns-Masur-Wilkinson paper).

*Proof:* By differentiating , we see that

Here, we used the fact that the WP metric is Kähler, so that is *parallel* (“commutes with ”).

Now, we observe that, by Wolpert’s formulas (cf. Theorem 3), one can write and that

and

Since (by definition), we conclude from the previous equations that

This proves the lemma.

Remark 15This ODE is an analogue for the WP metric of Clairaut’s relation for the “model metric” on the surface of revolution of the profil .

Lemma 16 (Lemma 4.15 of Burns-Masur-Wilkinson paper)

*Proof:* By definition, , so that

Differentiating this equality and using Wolpert’s formulas (Theorem 3), we see that

(Here, we used in the first equality the fact that is a geodesic, i.e., .)

It follows that

This proves the lemma.

]]>

This research announcement has 6 pages and it is divided into two parts:

- a) we present a short (complete) proof of a
*polynomial*upper bound for the rate of mixing of Weil-Petersson (WP) flow on the unit cotangent bundles of moduli spaces of surfaces of genus and punctures with , and - b) we give a sketch of proof of the
*rapid*(i.e., faster than polynomial) mixing property for the Weil-Petersson (WP) flow on the unit cotangent bundles of moduli spaces of four-times punctered spheres and once-punctured torii (that is, in the cases when ).

As we explain in our note, the speed of mixing of the WP flow on the unit cotangent bundle of is polynomial *at most* when because the (strata of the) boundary of the moduli spaces (in Deligne-Mumford compactification) looks like a *non-trivial* product of WP metrics on *non-trivial* moduli spaces of surfaces of lower “complexity” (smaller genus and/or less punctures) when .

More concretely, using this geometrical information on the WP metric near the boundary of , one can produce *lots* of geodesics spending a *lot of time* near the boundary of the moduli spaces traveling *almost-parallel* to one of the factors of the products of “lower complexity boundary moduli spaces” (so that the neighborhoods of the moduli spaces take a long time to see the compact parts, and, a fortiori, the rates of mixing between the compact parts and neighborhoods of the boundary are not very fast). Formally, for each , one can produce a subset of volume of vectors leading to WP geodesics traveling in the -thin part of (= -neighborhood of the boundary of ) for a time . In other words, there is a subset of volume of geodesics taking time *at least* to visit the -*thick* part of (= complement of the -thin part of ). Hence, there are certain *non-negligible* (volume ) subsets of the -thin part of taking longer and longer (time at least) to mix with the -thick part of as , so that the rate of mixing of the WP flow can not be very fast (i.e., the rate of mixing is not exponential, and, actually, not even a high degree polynomial).

This argument is not hard to formalize once one dispose of adequate estimates on the geometry of the WP metric near the boundary of (for instance, it occupies just 1 page of our note), and we will see it again in this blog in the last post of our series “*Dynamics of the Weil-Petersson flow*”.

In particular, it is not surprising that we were aware of this argument since 2010: in fact, as far as I can remember, this argument showed up in one of the discussions we had during the intervals of the talks of this conference in honor of Wolpert’s 60th birthday.

On the other hand, this argument breaks down in the case of the moduli spaces of four-times punctured spheres and once-punctured torii because the boundary of these particular moduli spaces consists of a single-point (and, thus, it is impossible to travel “almost-parallel” to the boundary in these particular cases).

This leads us to item b) above: as we announce in our note, the WP flow on the unit cotangent bundles of the particular moduli spaces and is actually *rapid* (faster than any given polynomial) mixing.

Geometrically, this is intuitively explained by the fact that the WP metric near the boundary of and looks like the metric of the surface of revolution of the profile . In particular, one can check that *any *WP-geodesic not going straight into the cusp (i.e., hitting the single boundary point of these particular moduli spaces) comes back to the -thick part in time . Since the WP metric has uniformly bounded negative curvature in the -thick part and the geodesic flows on negatively curved surfaces tend to be exponentially mixing, it is reasonable to expect rapid mixing (and maybe even exponential mixing) in these particular cases.

Nevertheless, the implementation of this idea is *technically* subtle because the curvature of the WP metric near the boundary point of and converges to so that the standard dynamical tools (such as “bounded distortion”) can not be employed directly.

As the reader can imagine, the previous paragraph partly explains why we are taking our time to write down our article.

This being said, the reason why we decided to release this research announcement note is now more or less clear. Indeed, after the 2010 paper of Burns, Masur and Wilkinson on the ergodicity of the WP flow, it is natural to ask about rates of mixing of this flow, and, for instance, several colleagues asked us about this property during a workshop on the WP metric in 2012 at Palo Alto. Here, while the argument for the proof of the result in item a) is extremely simple (and it is written since 2010), we kept (more or less) silent about this project until now *only* because we thought that the details of item b) could be filled out before the end of 2013 (thus allowing us to upload to ArXiv a more complete paper). Of course, since it is now clear that our initial plan of filling in the details of item b) before Christmas 2013 was too optimistic (on one hand) and it is not reasonable to write a paper *just* with the simple argument showing item a) (on the other hand), we decided that a research announcement note was the best solution to inform our friends of the results on the rates of mixing of the WP flow that we could get so far.

Closing this short post, let us point out that the end of our note contains a remark that item a) above gives a polynomial upper bound on the rates of mixing of the WP flow but no polynomial *lower *bound for these rates. Heuristically speaking, we think that one of the difficulties in showing such lower bounds is the absence of precise asymptotic estimates on how the curvatures of the WP metric approach zero (because such estimates would control how *close* to a product metric is the WP metric near the strata of the boundary of the moduli spaces, and, thus for how long can a geodesic travel almost-parallel to the strata of the boundary of the moduli spaces).

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In particular, this will “reduce” the proof of the Burns-Masur-Wilkinson theorem (of ergodicity of WP flow) to the verification of the last three items of Burns-Masur-Wilkinson ergodicity criterion for the WP metric and the proof of the Burns-Masur-Wilkinson ergodicity criterion itself.

We organize this post as follows. In next section we will quickly review some basic features of the moduli spaces of curves. Then, in the subsequent section, we will start by recalling the relationship between quadratic differentials on Riemann surfaces and the cotangent bundle of the moduli spaces of curves. After that, we will introduce the Weil-Petersson and the Teichmüller metrics. Finally, the last section of this post will concern the verification of the first three items of the Burns-Masur-Wilkinson ergodicity criterion for the WP metric.

The basic reference for the next two sections is Hubbard’s book.

**1. Moduli spaces of curves **

** 1.1. Definition and examples of moduli spaces **

Let be a fixed topological surface of genus with punctures. The *moduli space* is the set of Riemann surface structures on *modulo* biholomorphisms (conformal equivalences).

Example 1 (Moduli space of triply punctured spheres)The moduli space of triply punctured spheres consists of a single point

where denotes the Riemann sphere. Indeed, this is a consequence of the fact/exercise that the group of biholomorphisms (Möbius transformations) of the Riemann sphere issimply 3-transitive, i.e., given points , there exists an unique biholomorphism of sending , and (resp.) to , and (resp.).

Example 2 (Moduli space of once punctured torii)The moduli space of once punctured torii is

where the group acts on the hyperbolic half-plane via Möbius transformations (i.e., acts on via ). Indeed, this fact (previously explained in this post here) follows from the facts that a complex torus with a marked point is biholomorphic to a “normalized” lattice for some (with the marked point corresponding to the origin) and two “normalized” lattices are and are biholomorphic if and only if for some .

The second example reveals an interesting feature of : it is *not* a manifold, but only an *orbifold*. In fact, the *stabilizer* of the action of on at a *typical* point is trivial, but it has order at and order at (this corresponds to the fact that a typical torus has no symmetry, but the square and hexagonal torii have some symmetries). In particular, is topologically an once punctured sphere with two conical singularities at and . For a classical fundamental domain of the action of on , see the figure below (and also this one in the Wikipedia article on the modular group)

As it turns out, all moduli spaces are complex orbifolds. In order to see this fact, we need to introduce some auxiliary structures (including the notions of *Teichmüller spaces* and *mapping class groups*).

Remark 1From now on, we will restrict our attention to the case of a topological surface of genus with punctures such that . In this case, the uniformization theorem says that a Riemann surface structure on is conformally equivalent to a quotient of the hyperbolic upper-half plane by a discrete subgroup of (isomorphic to the fundamental group of ). Moreover, the hyperbolic metric on descends to a finite area hyperbolic metric on and, in fact, is the unique Riemannian metric of constant curvature on inducing thesameconformal structure. (See, e.g., Hubbard’s book for more details)

** 1.2. Teichmüller metric **

Let us start by endowing the moduli spaces with the structure of complete metric spaces.

By definition, a metric on corresponds to a way to “compare” (measure the distance) between two distinct points in the moduli space . A natural way of telling how far apart are two conformal structures on is by the means of quasiconformal maps.

*Very roughly speaking*, the idea is that even though by definition there is no conformal maps (biholomorphisms) between conformal structures and corresponding two distinct points of , one has several *quasiconformal maps* between them, that is, such that the quantity

is *finite*.

Here, it is worth to point out that is measuring the largest possible eccentricity among all infinitesimal ellipses in the tangent planes obtained as images under of infinitesimal circles on the tangent planes (see this post here or Hubbard’s book for more details [including some pictures of the geometrical meaning of ]), and, moreover, is conformal if and only if .

This motivates the following way of measuring the “distance” between and :

This function is the so-called *Teichmüller metric* (because, of course, it can be shown that is a metric on ).

The moduli space endowed with is a *complete* metric space.

Example 3The Teichmüller metric on the moduli space of once-punctured torii can be shown to coincide with the hyperbolic metric induced by Poincarés metric on (see Hubbard’s book).

** 1.3. Teichmüller spaces and mapping class groups **

Once we know that the moduli spaces are topological spaces (and, actually, complete metric spaces), we can start the discussion of its universal cover.

In this direction, we need to describe the “fiber” in the universal cover of a point of (i.e., a complex (Riemann surface) structure on ). In other terms, we need to add “extra information” to . As it turns out, this “extra information” has topological nature and it is called a *marking*.

More precisely, a *marked complex structure* (on ) is the data of a Riemann surface together with a homeomorphism (called marking).

By analogy with the notion of moduli spaces, we define the *Teichmüller space* is the set of Teichmüller equivalence classes of marked complex structures, where two marked complex structures and are *Teichmüller equivalent* whenever there exists a conformal map isotopic to . In other words, the Teichmüller space is the “moduli space of marked complex structures”.

The Teichmüller metric also makes sense on the Teichmüller space and the metric space is also complete.

From the definitions, we see that one can recover the moduli space from the Teichmüller space by forgetting the “extra information” given by the markings. Equivalently, we have that where is the so-called *mapping class group* of isotopy classes of orientation-preserving homeomorphisms of .

The mapping class group is a discrete group acting on by isometries of the Teichmüller metric . Moreover, by Hurwitz theorem (and the fact that we are assuming that ), the -stabilizer of any point of is finite (of cardinality when ), but it might vary from point to point because some Riemann surfaces are more symmetric than others (see, e.g., Example 2 above).

Example 4The Teichmüller space of once-punctured torii is . Indeed, the set of once-punctured torii is parametrized by normalized lattices , , and there is a conformal map between and if and only if , . Now, using this information one can check that and (because the conformal map associated to is isotopic to the identity if and only if ).

The Teichmüller space is the universal cover of and is the (orbifold) fundamental group of (compare with the example above). A common way to see this fact passes through showing that is simply connected (and even contractible) because it admits a *global* system of coordinates called Fenchel-Nielsen coordinates (providing an homemorphism between and ). The discussion of these coordinates is the topic of the next subsection.

** 1.4. Fenchel-Nielsen coordinates **

In order to introduce the Fenchel-Nielsen coordinates, we need the notion of *pants decomposition*. A pants (trouser) decomposition of is a collection of simple closed curves on that are pairwise disjoint, homotopically non-trivial (i.e., not homotopic to a point) and *non-peripheral* (i.e., not homotopic to a small loop around one of the possible punctures of ). The picture below illustrates a pants decomposition of a compact surface of genus :

The nomenclature “pants decomposition” comes from the fact that if we cut along the curves , (i.e., we consider the connected components of the complement of these curves), then we see “pairs of pants” (topologically equivalent to a triply punctured sphere):

A remarkable fact about pair of pants/trousers is that hyperbolic/conformal structures on them are *uniquely* determined by the (hyperbolic) lengths of their boundary components. In other terms, a trouser with boundary circles ( or ) has a -dimensional space of hyperbolic structures (parametrized by the lenghts of these -circles). Alternatively, one can construct trousers out of right-angled hexagons in the hyperbolic plane (see, e.g., Theorem 3.5.8 in Hubbard’s book).

In this setting, the Fenchel-Nielsen coordinates can be described as follows. We fix a pants decomposition and we consider

defined by , where is the hyperbolic length of with respect to the hyperbolic structure associated to the marked complex structure , and is a *twist parameter* measuring the “relative displacement” of the pairs of pants glued at .

A detailed description of the twist parameters can be found in Section 7.6 of Hubbard’s book, but, for now, let us just make some quick comments about them. Firstly, we fix (in an *arbitrary* way) a collection of simple arcs joining the boundaries of the pairs of pants determined by such that these arcs land at the same point whenever they come from opposite sides of .

From these arcs, we get a collection of simple closed curves on looking like this:

Consider now a pair of trousers sharing a curve (they might be the same trouser) and let be an arc of a curve in joining two boundary components and of the union of these trousers:

Given a marked complex structure , consider the unique arc on homotopic to (relative to the boundary of the union of the pair of trousers) consisting of two *minimal* geodesic arcs connecting to and and an immersed geodesic moving inside . We define the twist parameter as the oriented length of counted as positive if it turns to the right and negative if it turns to the left. The figure below illustrates two markings and whose twist parameters differ by

Remark 2The fact that the definition of the twist parameters depend on the choice of implies that the twist parameters are well-defined only up to a constant. Nevertheless, this technical difficulty does not lead to any serious issue.

In any case, it is possible to show the Fenchel-Nielsen coordinates associated to any pants decomposition is a *global* homeomorphism (see, e.g., Theorem 7.6.3 in Hubbard’s book). In particular, the Teichmüller space is simply connected (as it is homeomorphic to ). Hence, it is the universal cover of the moduli space (and the mapping class group is the orbifold fundamental group of ).

This partly explain why one discusses the properties of and at the same time.

**2. Cotangent bundle of moduli spaces **

Another reason for studying and together is because is a manifold while is only an orbifold. In fact, the Teichmüller spaces are *real-analytic manifolds*. Indeed, the real-analytic structure on comes from the uniformization theorem. More precisely, given a marked complex structure , we can apply the uniformization theorem to write where is a discrete subgroup isomorphic to the fundamental group of . In other words, from a marked complex structure , we have a representation of on (well-defined *modulo conjugation*), and this permits to identify with an open component of the character variety of homomorphisms from to modulo conjugacy. In particular, the pullback of the real-analytic structure of this representation variety to endow with its own real-analytic structure.

Actually, as it turns out, this real-analytic structure of can be “upgraded” to a *complex-analytic structure*. One way of seeing this uses a “generalization” of the construction of the real-analytic structure above based on the complex-analytic structure on the representation variety of in and Bers simultaneous uniformization theorem.

Remark 3Let be a real vector space of dimension and denote by the set of linear complex structures on (i.e., -linear maps with ). It is possible to check that a linear complex structure on isIt is worth to compare this with the following “toy model” situation.equivalentto the data of a complex subspace of the complexification of such that and (i.e., ) where is the complex conjugate of .

Since the condition isopenin the Grassmanian manifold of complex subspaces of of complex dimension , and is naturally a complex manifold, we obtain that the set parametrizing complex structures on is itself a complex manifold.

We will discuss this point later (in a future post) and, for now, let us just sketch the relationship between the *quadratic differentials* on Riemann surfaces and the cotangent bundle to Teichmüller and moduli spaces (referring to this previous post for more details).

** 2.1. Integrable quadratic differentials **

The Teichmüller metric was defined via the notion of quasiconformal mappings . By inspecting the nature of this notion, we see that the quantities (related to the eccentricities of infinitesimal ellipses obtained as the images under of infinitesimal circles) play an important role in the definition of the Teichmüller distance between and .

The measurable Riemann mapping theorem of Alhfors and Bers (see, e.g., page 149 of Hubbard’s book) says that the quasiconformal map can be *recovered* from the quantities *up to composition with conformal maps*. More precisely, by collecting the quantities in a globally defined *tensor* of type

with called *Beltrami differential*, one can “recover” by solving *Beltrami’s equation*

in the sense that there is always a solution to ths equation and, furthermore, two solutions and differ by a conformal map (i.e., ).

In other terms, the deformations of complex structures are intimately related to Beltrami differentials and it is not surprising that Beltrami differentials can be used to describe the tangent bundle of . In this setting, we can obtain the cotangent bundle by noticing that there is a natural pairing between bounded () Beltrami differentials and integrable () *quadratic differentials* (i.e., a tensor of type , ):

because is an area form and is integrable. In this way, it can be shown that the cotangent space at a point of is naturally identified to the space of integrable quadratic differentials on .

Note that the space of integrable quadratic differentials provides a concrete way of manipulating the complex structure of : in this setting, the complex structure is just the multiplication by on the space of quadratic differentials.

Remark 4By a theorem of Royden (see Hubbard’s book), the mapping class group is the group of complex-analytic automorphisms of . In particular, the moduli space is a complex orbifold.

** 2.2. Teichmüller and Weil-Petersson metrics **

Using the description of the cotangent bundle of in terms of quadratic differentials, we are ready to define the Teichmüller and Weil-Petersson metrics.

Given a point of , we endow the cotangent space with the -norm:

where is the hyperbolic metric associated to the conformal structure and is a quadratic differential (i.e., a tensor of type ).

Remark 5More generally, we define the -norm of a tensor of type (i.e., ) as:

In this notation, the *infinitesimal Teichmüller metric* is the family of -norms on the fibers of the cotangent bundle of . Here, the nomenclature “infinitesimal Teichmüller metric” is justified by the fact that the “global” Teichmüller metric (defined by the infimum of the eccentricity factors of quasiconformal maps ) is the Finsler metric induced by the “infinitesimal” Teichmüller metric (see, e.g., Theorem 6.6.5 of Hubbard’s book).

In a similar vein, the *Weil-Petersson (WP) metric* is the family of -norms on the fibers of the cotangent bundle of .

Remark 6In the definition of the Weil-Petersson metric, it was implicit that an integrable quadratic differential has finite -norm (and, actually, all -norms are finite, ). This fact is obvious when the is compact, but it requires a (simple) computation when has punctures. See, e.g., Proposition 5.4.3 of Hubbard’s book for the details.

For later use, we will denote the (infinitesimal) Teichmüller metric, resp., Weil-Petersson metric, as , resp. .

The Teichmüller metric is a Finsler metric in the sense that the family of -norms on the fibers of vary in a but not way (cf. Lemma 7.4.3 and Proposition 7.4.4 in Hubbard’s book).

Remark 7The first derivative of the Teichmüller metric is not hard to compute. Given two cotangent vectors with , we affirm that

Indeed, the first derivative is . Since and is bounded (i.e., its norm is finite), we can use the dominated convergence theorem to obtain that

The Weil-Petersson metric is induced by the Hermitian inner product

As usual, the real part induces a real inner product (also inducing the Weil-Petersson metric), while the imaginary part induces an anti-symmetric bilinear form, i.e., a symplectic form.

By definition, the Weil-Petersson metric relates to the Weil-Petersson symplectic form and the complex structure on (i.e., multiplication by of elements of ) via:

Furthermore, as it was firstly discovered by Weil by means of a “simple-minded calculation” (“*calcul idiot*”) and later confirmed by others, it is possible to show that the Weil-Petersson metric is Kähler, i.e., the Weil-Petersson symplectic form is closed (that is, its exterior derivative vanishes: ). See, e.g., Section 7.7 of Hubbard’s book for more details.

We will come back later (in a future post) to the Kähler property of the Weil-Petersson metric, but for now let us just mention that this property enters into the proof of a beautiful theorem of Wolpert saying that the Weil-Petersson symplectic form has a simple expression in terms of Fenchel-Nielsen coordinates:

where is an arbitrary pants decomposition of . Here, it is worth to mention that an important step in the proof of this formula (cf. Step 2 in the proof of Theorem 7.8.1 in Hubbard’s book) is the fact discovered by Wolpert that the infinitesimal generator of the Dehn twist about is of the symplectic gradient of the Hamiltonian function , that is,

This equation is the starting point of several Wolpert’s expansion formulas for the Weil-Petersson metric that we will discuss later in this series of posts.

Before proceeding further, let us briefly discuss the Teichmüller and Weil-Petersson metrics on the moduli spaces of once-punctured torii .

Example 5On the other hand, the Fenchel-Nielsen coordinates on have first-order expansionThe Teichmüller metric on is the quotient of the hyperbolic metric of .

where . Thus, we see from Wolpert’s formula thatSince the complex structure on is the standard complex structure of , we see that the Weil-Petersson metric has asymptotic expansion

that is, the Weil-Petersson on the moduli space near the cusp at infinity is modeled by the surface of revolution obtained by rotating the curve (for say). See the picture below. This is in contrast with the fact that the Teichmüller metric is the hyperbolic metric and hence it is modeled by surface of revolution obtained by rotation the curve (for say). (Recall that, in general, a surface of revolution obtained by rotation of the curve has the metric )From this asymptotic expansion of , we see that it isincomplete: indeed, a vertical ray to the cusp at infinity starting at a point in the line has Weil-Petersson length . Moreover, the curvature satisfies , and, in particular, as .

The previous example (Weil-Petersson metric on ) already contains several features of the Weil-Petersson metric on *general* Teichmüller spaces and moduli spaces .

In fact, we will see later that the Weil-Petersson metric is incomplete because it is possible to shrink a simple closed curve to a point and leave Teichmüller space along a Weil-Petersson geodesic in time . Also, some sectional curvatures might approach as one leaves Teichmüller space.

Nevertheless, an interesting feature of the Weil-Petersson metric in and for *not* occuring in the case of is the fact that some sectional curvatures might also approach as one leaves Teichmüller space. Indeed, as we will see later, this happens because the “boundary” of is sufficiently “large” when so that it is possible form some Weil-Petersson geodesics to travel “almost parallel” to certain parts of the “boundary” for a certain time (while the same is *not* possible for because the “boundary” consists of a single point).

After this brief introduction of our main dynamical object (Weil-Petersson geodesic flow), we can now start the discussion of the proof of Burns-Masur-Wilkinson theorem (on the ergodicity of the Weil-Petersson flow). The basic reference for the next two sections is Burns-Masur-Wilkinson paper.

**3. Burns-Masur-Wilkinson theorem and ergodicity of the Weil-Petersson flow on finite covers of moduli spaces **

Recall that the statement of Burns-Masur-Wilkinson ergodicity criterion for geodesic flows on *manifolds* is:

Theorem 1 (Burns-Masur-Wilkinson)

geodesically convex, i.e., for every , there exists an unique geodesic segmentinconnecting and .- (II) the metric completion of is
compact.volumetrically cusplike, i.e., for some constants and , the volume of a -neighborhood of the boundary satisfiesfor every .

polynomially controlled curvature, i.e., there are constants and such that the curvature tensor of and its first two derivatives satisfy the following polynomial boundfor every .

polynomially controlled injectivity radius, i.e., there are constants and such thatfor every (where denotes the injectivity radius at ).

first derivative of the geodesic flowispolynomially controlled, i.e., there are constants and such that, for every infinite geodesic on and every :non-uniformly hyperbolic(in the sense of Pesin’s theory) andergodic.

The goal of this section is to show how the ergodicity criterion above can be used to deduce the following theorem (Burns-Masur-Wilkinson theorem on the ergodicity of the Weil-Petersson geodesic flow).

Theorem 2 (Burns-Masur-Wilkinson)The Weil-Petersson flow on the unit cotangent bundle of is ergodic (for any , ) with respect to the Liouville measure of the WP metric. Actually, it is Bernoulli (i.e., it is measurably isomorphic to a Bernoulli shift) and,a fortiori, mixing. Furthermore, its metric entropy is positive and finite.

At first sight, it is tempting to say that Theorem 2 follows from Theorem 1 after checking items (I) to (VI) for the case (the cotangent bundle of ), (the cotangent bundle of ) and (the mapping class group).

However, this is not quite true because the moduli spaces and their unit cotangent bundles are not *manifolds* but only *orbifolds*, while we assumed in Theorem 1 that the phase space of the geodesic flow is a manifold.

In other words, the orbifoldic nature of moduli spaces imposes a technical difficulty in the reduction of Theorem 2 to Theorem 1. Fortunately, a solution to this technical issue is very well-known to algebraic geometers and it consists into taking an adequate *finite* cover of the moduli space in order to “kill” the orbifold points (i.e., points with large stabilizers for the mapping class group).

More precisely, for each , one considers the following *finite-index* subgroup of the mapping class group :

where is the action on homology of . Equivalently, an element of belongs to whenever its action on the absolute homology group corresponds to a (symplectic) integral matrix congruent to the identity matrix modulo .

Example 6In the case of once-punctured torii, the mapping class group is and

In the literature, is called the principal congruence subgroup of of level .

Remark 8The index of in can be computed explicitly. For instance, the natural map from to is surjective (see, e.g., Farb-Margalit’s book), so that the index of is the cardinality of , and, for prime, one has

cf. Dickson’s book.

It was shown by Serre (see the appendix of this paper here for the original proof and the Chapter 6 of the book of Farb-Margalit for an alternative exposition) that is torsion-free for and, *a fortiori*, it acts freely and properly discontinuous on for . In other terms, the finite cover of given by

is a *manifold* for .

Remark 9Serre’s result is sharp: the principal congruence subgroup of level of contains the torsion element .

Once one disposes of an appropriate manifold finitely covering the moduli space , the reduction of Theorem 2 to Theorem 1 consists into two steps:

- (a) the verification of items (I) to (VI) in the statement of Theorem 1 in the case of the unit cotangent bundle of .
- (b) the deduction of the ergodicity (and mixing, Bernoullicity, and positivity and finiteness of metric entropy) of the Weil-Petersson geodesic flow on from the corresponding fact(s) for the Weil-Petersson geodesic flow on .

For the remainder of this section, we will discuss item (b) while leaving the first part of item (a) (i.e., items (I), (II) and (III) of Theorem 1 for ) for the next section and the second part of item (a) (i.e., items (I), (II) and (III) of Theorem 1 for )) for the next post.

For ease of notation, we will denote , and . Assuming that the Weil-Petersson flow is ergodic (and Bernoulli, and its metric entropy is positive and finite) on , the “obstruction” to show the same fact(s) for the Weil-Petersson flow on is the possibility that the orbifold points of form a “large” set.

Indeed, if we can show that the set of orbifold points of is “small” (e.g., they form a set of zero measure), then the geodesic flow on covers the geodesic flow on on a set of full measure. In particular, if is a (Weil-Petersson flow) invariant set of positive measure on , then its lift to is also a (Weil-Petersson flow) invariant set of positive measure. Therefore, by the ergodicity of the Weil-Petersson flow on , we have that has full measure, and, *a fortiori*, has full measure. Moreover, the fact that the Weil-Petersson flow on covers the Weil-Petersson flow on on a full measure set also allows to deduce Bernoullicity and positivity and finiteness of metric entropy of the latter flow from the corresponding properties for the former flow.

At this point, it remains only to check that the orbifold points of for a subset of zero measure (for the Liouville/volume measure of the Weil-Petersson metric) in order to complete the discussion of this section. In this direction, we have that the following fact:

Lemma 3Let be the subset of corresponding to orbifoldic points, i.e., is the (countable) union of the subsets of fixed points of the natural action on of all elements of finite order,excludingthe genus hyperelliptic involution. Then, is a closed subset of real codimension (at least).

*Proof:* For each of finite order, is the Teichmüller space of the quotient orbifold . From this, one can show that:

- if is compact and is not the hyperelliptic involution in genus , then has complex dimension ;
- if has punctures, then has complex dimension ;
- if is the hyperelliptic involution in genus , then .

See, e.g., this paper of Rauch for more details.

In particular, the proof of the lemma is complete once we verify that is a *locally finite* union of the real codimension subsets , .

Keeping this goal in mind, we fix a compact subset of and we recall that the mapping class group acts in a properly discontinuous manner on . Therefore, it is not possible for an infinite sequence of distinct finite order elements to satisfy for all . In other words, is the subset of finitely many , i.e., is a locally finite union of , .

Example 7In the case of once-punctured torii, the subset consists of the -orbits of the points and .

**4. Geometry of the Weil-Petersson metric: part I **

In this section, we will discuss three properties of the Weil-Petersson metric on and related to the items (I), (II) and (III) in the statement of Theorem 1 above.

We start by noticing that the item (I) in the statement of Theorem 1 (i.e., the geodesic convexity of the Weil-Petersson metric on ) was proved by Wolpert in this paper here.

Next, the fact that it is possible to leave the Teichmüller space along a Weil-Petersson geodesic in finite time of order was exploited by Masur to show that the metric completion of the (equipped with the Weil-Petersson metric) is the so-called *augmented Teichmüller space* . As it turns out, the mapping class group acts on and the quotient

is the so-called Deligne-Mumford compactification of the moduli space (giving the metric completion of equipped with the Weil-Petersson metric).

The augmented Teichmüller space is a stratified space obtained by adjoining lower-dimensional Teichmüller spaces of *noded* Riemann surfaces. The combinatorial structure of the stratification of is encoded by the *curve complex* (also called *complex of curves* or graph of curves in the literature).

More precisely, the curve complex is a -simplicial complex defined as follows. The vertices of are homotopy classes of homotopically non-trivial, non-peripheral, simple closed curves on . We put an edge between two vertices whenever the corresponding homotopy classes have *disjoint* representatives. In general, a -simplex consists of distinct vertices possessing mutually disjoint representatives.

Remark 10is a -simplicial complex because a maximal collection of distinct vertices possessing disjoint representatives is a pants decomposition of and, hence, .

Example 8In the case of once-punctured torii, the curve complex consists of an infinite discrete set of vertices (because there is no pair of disjoint homotopically distinct curves). However, some authors define the curve complex of once-punctured torii by putting an edge between vertices corresponding to curves intersecting minimally (i.e., only once). In this definition, the curve complex of once-punctured torii becomes the Farey graph.

The curve complex is a connected locally infinite complex, except for the cases of the four-times punctured spheres and the once-punctured torii. Also, the mapping class group acts on . Moreover, Masur and Minsky showed that is -hyperbolic metric space for some .

Using the curve complex , we can define the augmented Teichmüller space as follows.

A *noded Riemann surface* is a compact topological surface equipped with the structure of a complex space with at most isolated singularities called *nodes* such that each of these singularities possess a neighborhood biholomorphic to a neighborhood of in the singular curve

Removing the nodes of a noded Riemann surface yields to a possibly disconnected Riemann surface denoted by . The connected components of are called the *pieces* of . For example, the noded Riemann surface of genus of the figure above has two pieces (of genera and 1 resp.).

Given a simplex , we will adjoint a Teichmüller space to in the following way. A *marked noded Riemann surface* with nodes at is a noded Riemann surface equipped with a continuous map such that the restriction of to is a homeomorphism to . We say that two marked noded Riemann surfaces and are *Teichmüller equivalent* if there exists a biholomorphic node-preserving map such that is isotopic to . The Teichmüller space associated to is the set of Teichmüller equivalence classes marked noded Riemann surfaces with nodes at .

In this context, the augmented Teichmüller space is

The topology on is given by the following neighborhoods of points . Given , we consider a maximal simplex (pants decomposition of ) containing and we let be the corresponding Fenchel-Nielsen coordinates on . We *extend* these coordinates by allowing whenever is pinched in a node and we take the quotient by identifying noded Riemann surfaces corresponding to parameters and whenever .

Remark 11The augmented Teichmüller space is not locally compact: indeed, a neighborhood of a noded Riemann surface allows for arbitrary twists corresponding to curves .

The quotient of by the natural action of (through the corresponding action on ) is called *Deligne-Mumford compactification* of the moduli space (see, e.g., this paper of Hubbard and Koch for more details). Since is a finite-index subgroup of and is the metric completion of with respect to the Weil-Petersson metric, it follows that the the metric completion of with respect to the Weil-Petersson metric is also compact.

In particular, satisfies the item (II) in the statement of Theorem 1.

Remark 12It is worth to notice that the Deligne-Mumford compactification in the case of the once-punctured torii is just one point (because geometrically by pinching one curve in a punctured torus we get a thrice-puncture sphere in the lmit) while it is stratified in non-trivial lower-dimensional moduli spaces in general. Moreover, as we will see later, some asymptotic formulas of Wolpert tells that the Weil-Petersson metric “looks” like a product of the Weil-Petersson metrics on these lower-dimensional moduli spaces.In particular, as we will discuss in the last post of this series, it will be possible for several Weil-Petersson geodesics to travel “almost parallel” to these lower-dimensional moduli spaces and this will give a polynomial rate of mixing for this flow in general. On the other hand, since it is not possible to travel almost parallel to a point for a long time, this arguments breaks down in the case of the Weil-Petersson metric in the case of the moduli space of once-punctured torii.

Finally, let us complete the discussion in this section by quickly checking that also satisfies the item (III) in the statement of Theorem 1, i.e., its boundary is volumetrically cusp-like.

In this direction, given , let us denote by the Weil-Petersson distance between and . Our current task is to prove that there are constants and such that

where .

As we are going to see now, one can actually take in the estimate above thanks to some asymptotic formulas of Wolpert for the Weil-Petersson metric near .

Lemma 4One has .

*Proof:* It was shown by Wolpert (in page 284 of this paper here) that the Weil-Petersson metric has asymptotic expansion

near , where and , are the Fenchel-Nielsen coordinates associated to .

This gives that the volume element of the Weil-Petersson metric near is . Furthermore, this also says that the distance between and is comparable to . By putting these two facts together, we see that

This proves the lemma.

Remark 13In a recent work, Mirzakhani studied the total mass of with respect to the Weil-Petersson metric and she showed that there exists a constant such thatUsing the properties that the metric completion of is compact and is volumetrically cusp-like imply that the Liouville measure (volume) is finite.

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This event is part of the activities around the chaire Jean-Morlet of Boris Hasselblatt. Among the topics scheduled in this event, there is a mini-course by Keith Burns and myself around the dynamics of the Weil-Petersson (WP) geodesic flow.

In our mini-course, Keith and I plan to cover some aspects of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow and, maybe, some points of our joint work with Masur and Wilkinson on the rates of mixing of WP flow.

In order to help me prepare my talks, I thought it could be a good idea to make my notes available on this blog.

So, this post starts a series of 6 posts (vaguely corresponding the 6 lectures of the mini-course) on the dynamics of the WP flow.

The Weil-Petersson flow (WP flow) is a certain geodesic flow (of the Weil-Petersson metric) on the unit cotangent bundle of the moduli space of curves (Riemann surfaces) of genus with marked points.

The WP flow and its close cousin the *Teichmüller flow* are studied in the literature in part because its dynamical properties allow to understand certain geometrical aspects of Riemann surfaces.

The precise definitions of these flows will be given later, but, for now, let us list some of their properties.

Teichmüller flow | WP flow | |

(a) | comes from a Finsler | comes from a Riemannian metric |

(b) | complete | incomplete |

(c) | is part of an action | is not part of an action |

(d) | non-uniformly hyperbolic | singular hyperbolic |

(e) | related to flat geometry of curves | related to hyperbolic geometry of curves |

(f) | transitive | transitive |

(g) | periodic orbits are dense | periodic orbits are dense |

(h) | finite topological entropy | infinite topological entropy |

(i) | ergodic for the Liouville measure | ergodic for the Liouville measure |

(j) | metric entropy | metric entropy |

(k) | exponential rate of mixing | mixing at most polynomial (in genus ) |

The items above serve to highlight some differences between the Teichmüller and WP flows.

In fact, the Teichmüller flow is associated to a Finsler, i.e., a continuous family of norms, on the fibers of the cotangent bundle of