Last time, we showed the first part of Burns-Masur-Wilkinson ergodicity criterion:

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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