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.

]]>

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.

]]>

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).

]]>

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)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).

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.

]]>

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 the moduli spaces (actually, a but not family of norms [see pages 308 and 309 of Hubbard's book]), while the WP flow is associated to a Riemannian and, actually, Kähler, metric. We will come back to this point later when defining the WP metric.

In particular, the item (a) says that the WP flow comes from a metric that is richer than the metric generating the Teichmüller flow.

On the other hand, the item (b) says that the WP geodesic flow has a not so nice dynamics because it is *incomplete*, that is, there are certain WP geodesics that “leave”/“go to infinity” in *finite* time. In particular, the WP flow is *not* defined for all time when we start from *certain* initial datum. We will make more comments on this later. Nevertheless, Wolpert showed that the WP flow is defined for all time for *almost every* initial data (with respect to the Liouville [volume] measure induced by WP metric), and, thus, the WP flow is a legitim flow from the point of view of Dynamics/Ergodic Theory.

The item (c) says that WP flow differs from Teichmüller flow because the former is not part of a -action while the latter corresponds to the action of the diagonal subgroup of a natural -action on the unit cotangent bundle of the moduli spaces of curves. Here, it is worth to mention that the mere fact that the Teichmüller flow is part of a -action makes its dynamics very rich: for instance, once one shows that the Teichmüller flow is ergodic (with respect to some -invariant probability measure), it is possible to apply Howe-Moore’s theorem (or variants of it) to improve ergodicity into mixing (and, actually, exponential mixing) of Teichmüller flow (see e.g. this post for more details).

The item (d) says that both WP and Teichmüller flows are non-uniformly hyperbolic (in the sense of Pesin theory), but they are so for *distinct* reasons. The non-uniform hyperbolicity of the Teichmüller flow was shown by Veech (for the “volume”/Masur-Veech measure) and Forni (for an arbitrary Teichmüller flow invariant probability measure) and it follows from uniform estimates for the derivative of the Teichmüller flow on bounded sets. On the other hand, the non-uniform hyperbolicity of the WP flow requires a slightly different argument because the curvatures of WP metric might approach or at certain places near the “boundary” of the moduli spaces. We will return to this point in the future.

The item (e) says that, concerning applications of these flows to the investigation of curves/Riemann surfaces, it is natural to study the Teichmüller flow whenever one is interested in the properties of flat metrics with conical singularities (cf. this post here), while it is more natural to study the WP metric/flow whenever one is interested in the properties of hyperbolic metrics: for instance, Wolpert showed that the hyperbolic length of a closed geodesic in a fixed free homotopy class is a convex function along orbits of the WP flow, Mirzakhani proved that the growth of the hyperbolic lengths of simple geodesics on hyperbolic surfaces is related to the WP volume of the moduli space, and, after the works of Bridgeman, McMullen and more recently Bridgeman, Canary, Labourie and Sambarino (among other authors), we know that the Weil-Petersson metric is intimately related to *thermodynamical invariants* (entropy, pressure, etc.) of the geodesic flow on hyperbolic surfaces.

Concerning items (f) to (h), Pollicott-Weiss-Wolpert showed the transitivity and denseness of periodic orbits of the WP flow in the particular case of the unit cotangent bundle of the moduli space (of once-punctured tori). In general, the transitivity, the denseness of periodic orbits and the infinitude of the topological entropy of the WP flow on the unit cotangent bundle of (for any , ) were shown by Brock-Masur-Minsky. Moreover, Hamenstädt proved the *ergodic version* of the denseness of periodic orbits, i.e., the denseness of the subset of ergodic probability measures supported on periodic orbits in the set of all ergodic WP flow invariant probability measures.

The ergodicity of WP flow (mentioned in item (i)) was first studied by Pollicott-Weiss in the particular case of the unit cotangent bundle of the moduli space of once-punctured tori: they showed that *if* the first two derivatives of the WP flow on are suitably bounded, *then* this flow is ergodic. More recently, Burns-Masur-Wilkinson were able to control *in general* the first derivatives of WP flow and they used their estimates to show the following theorem:

Theorem 1 (Burns-Masur-Wilkinson)The WP 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.

A detailed explanation of this theorem will occupy the next four posts of this series. For now, we will just try to describe the general lines of Burns-Masur-Wilkinson arguments in Section 1 below.

However, before passing to this subject, let us make some comments about item (k) above on the rate of mixing of Teichmüller and WP flows.

Generally speaking, it is expected that the rate of mixing (decay of correlations) of a system (diffeomorphism or flow) displaying a “reasonable” amount of *hyperbolicity* is exponential: for example, the property of exponential rate of mixing was shown by Dolgopyat (see also this article of Liverani and this blog post) for contact Anosov flows (such as geodesic flows on compact Riemannian manifolds with negative curvature), and by Avila-Gouëzel-Yoccoz and Avila-Gouëzel for the Teichmüller flow equipped with “nice” measures.

Here, we recall that the *rate of mixing*/*decay of correlations* of a (mixing) flow is the speed of convergence of the correlations functions to as (for “reasonably smooth” observables and ), that is, the speed of to mix distinct regions of the phase space (such as the supports of the observables and ).

In this context, given the ergodicity and mixing theorem of Burns-Masur-Wilkinson, it is natural to try to “determine” the rate of mixing of WP flow. In this direction, we obtained the following result (in a preprint still in preparation):

Theorem 2 (Burns-Masur-M.-Wilkinson)The rate of mixing of WP flow on is

- at most
polynomialfor andrapid(super-polynomial) for , .

We will present a sketch of proof of this result in the last post of this series. For now, we will content ourselves with a vague description of the *geometrical reason* for the difference in the rate of mixing of the Teichmüller and WP flows in Section 2 below.

Closing this introduction, let us give a plan of this series of posts. Firstly, we will complete today’s post by discussing the general scheme for the proof of Burns-Masur-Wilkinson theorem (ergodicity of WP flow) in Section 1 below and by explaining the geometry behind the rate of mixing of WP flow in Section 2. Then, in the second post of this series, we will define the WP geodesic flow on the unit cotangent bundle of the moduli spaces of curves and we will “reduce” Burns-Masur-Wilkinson theorem to the verification of adequate estimates of the derivatives of WP flow via a certain *ergodicity criterion* à la Katok-Strelcyn. After that, we will spend the third and fourth post discussing the proof of the ergodicity criterion à la Katok-Strelcyn, and we will dedicate the fifth post to show that the WP geodesic flow satisfies all assumptions of the ergodicity criterion. Finally, the last post will concern the rates of mixing of WP flow.

**1. Ergodicity of WP flow: outline of proof **

The initial idea to prove Burns-Masur-Wlkinson theorem is the “usual” argument for the proof of ergodicity of a system exhibiting some hyperbolicity, namely, *Hopf’s argument*.

The general scheme of this argument is the following. Given a smooth flow on a compact Riemannian manifold preserving the corresponding volume measure and a continuous observable , we consider the future and past Birkhoff averages:

By Birkhoff’s ergodic theorem, for -almost every , the quantities and exist and, actually, they coincide . In the literature, a point such that , exist and is called a *Birkhoff generic* point (with respect to ).

By definition, the ergodicity of (with respect to ) is equivalent to the fact that the functions and are *constant* at -almost every point.

In order to show the ergodicity of a flow with some hyperbolicity, Hopf observes that the function , resp. , is constant along stable, resp. unstable, sets

i.e., whenever , resp. whenever . We leave the verification of this fact as an exercise to the reader.

In the case of a volume-preserving Anosov flow (sometimes called *uniformly hyperbolic flow*) on , we know that the stable and unstable sets are immersed *submanifolds*. Moreover, if one forgets about the flow direction, the stable and unstable manifolds have complementary dimensions and intersect transversely. Hence, given two points (lying in distinct orbits of ), we can connect them using pieces of stable and unstable manifolds as shown in the figure below:

In particular, this *indicates* that a volume-preserving Anosov flow is ergodic because the functions and are constant along stable and unstable manifolds, they coincide almost everywhere and any pair of points can be connected via pieces of stable and unstable manifolds. However, this argument towards ergodicity of is *not* complete yet: indeed, one needs to know that the intersection points between the pieces of stable and unstable manifolds connecting and are Birkhoff generic in order to conlude that .

In the original context of his article, Hopf studies a geodesic flow of a compact surface of constant negative curvature, and he uses the fact that the stable and unstable manifolds form foliations by curves to deduce that the intersection points can be taken to be Birkhoff generic points. Indeed, since the stable and unstable foliations are in his context, Hopf applies Fubini’s theorem to the set of full -volume consisting of Birkhoff generic points in order to ensure that almost all stable and unstable manifolds/curves and intersect in a subset of total length measure of and .

On the other hand, it is known that the stable and unstable manifolds of a general Anosov flow (such as geodesic flows on compact manifolds of variable negative curvature) do *not* form necessarily a -foliation, but only Hölder continuous foliations (see e.g. the papers of Anosov and/or Hasselblatt for concrete examples). In particular, this is an obstacle to the argument *à la Fubini* of the previous paragraph. Nevertheless, Anosov showed that the stable and unstable foliations of a smooth Anosov flow are always absolutely continuous, so that one can still apply Fubini’s theorem to conclude ergodicity along the lines of Hopf’s argument presented.

In summary, we know that a smooth () volume-preserving Anosov flow on a compact manifold is ergodic thanks to Hopf’s argument and the absolute continuity of stable and unstable foliations.

Remark 1Robinson-Young showed that the stable and unstable foliations of a Anosov system are not necessarily absolutely continuous. In particular, the smoothness () assumption on the Anosov flow is necessary for the ergodicity argument described above.

Remark 2The absolute continuity of a foliation invariant under some system depends on some hyperbolicity. In fact, Shub-Wilkinson constructed examples of invariant foliations of certain partially hyperbolic diffeomorphisms failing to satisfy Fubini’s theorem in the sense that each leaf of this foliation (along which the dynamics is central/neutral) intersects a set of full volume exactly at one point! This phenomenon is sometimes referred to as Fubini’s nightmare in the literature (see, e.g., this article of Milnor) and sometimes a foliation “failing” Fubini’s theorem is called a pathological foliation (see, e.g., these pictures by Wilkinson for some examples of such foliations)

After this brief sketch of Hopf’s argument for smooth volume-preserving Anosov flows on compact manifolds, let us explain the difficulties of extending this argument to the setting of WP flow.

As we already mentioned (cf. item (d) of the table above), the WP flow is singular hyperbolic. In a nutshell, this means that, even though WP flow is not uniformly hyperbolic (Anosov), it is non-uniformly hyperbolic in the sense of Pesin theory and it satisfies some hyperbolicity estimates along pieces of orbits staying in compact parts of moduli space.

In particular, thanks to Pesin’s stable manifold theorem, the stable and unstable sets of almost every point are immersed submanifolds, and, if we forget about the flow direction, the stable and unstable manifolds have complementary dimensions. Furthermore, the stable and unstable manifolds are part of absolutely continuous laminations. Here, it is important that the dynamics is sufficiently smooth (see, e.g., this paper of Pugh, and this preprint of Bonatti, Crovisier and Shinohara).

Thus, this gives hopes that Hopf’s argument *could be* applied to show the ergodicity of volume-preserving non-uniformly hyperbolic systems.

However, by inspecting the figure 1 above, we see that Hopf’s argument relies on the fact that stable and unstable manifolds of Anosov flows have a nice, well-controlled, geometry.

For instance, if we start with a point and we want to connect it with pieces of stable and unstable manifolds to a point at a *large* distance, we have to make sure that the pieces of stable and unstable manifolds used in figure 1 are “uniform”, e.g., they are graphs of *definite size* and *bounded curvature* with respect to the splitting into stable and unstable directions, and, moreover, the angles between the stable and unstable directions are *uniformly bounded away from zero*.

Indeed, if the pieces of stable and unstable manifolds get shorter and shorter, and/or if they “curve” a lot, and/or the angles between stable and unstable directions are not bounded away from zero, one might not be able to reach/access from with stable and unstable manifolds:

As it turns out, while these kinds of non-uniformity do not occur for Anosov flows, they can actually occur for *certain* non-uniformly hyperbolic systems. More precisely, the sizes and curvatures of stable and unstable manifolds, and the angles between stable and unstable directions of a general non-uniformly hyperbolic system vary only *measurably* from point to point.

In particular, this excludes *a priori* a naive generalization of Hopf’s ergodicity argument for non-uniformly hyperbolic systems, and, in fact, there are concrete examples by Dolgopyat, Hu and Pesin of volume-preserving non-uniformly hyperbolic systems with countably many ergodic components consisting of invariant sets of positive volumes that are essentially open (and, as a matter of fact, this example is “sharp” in the sense that Pugh and Shub showed that a volume-preserving non-uniformly hyperbolic system has at most countably many ergodic components).

In summary, the ergodicity of a non-uniformly hyperbolic system *depends* on the particular dynamical features of the given system.

In this direction, there is a *vast* literature dedicated to the construction of large classes of ergodic non-uniformly hyperbolic systems: for example, the ergodicity of several classes of billiards was shown by Sinai, Bunimovich, Chernov-Bunimovich-Sinai among others (see also the book of Chernov-Markarian) and the ergodicity of non-uniformly hyperbolic systems exhibiting some partial hyperbolicity (or some “dominated splitting”) was shown by Pugh-Shub, Rodriguez-Hertz, Tahzibi, Burns-Wilkinson, Rodriguez-Hertz– Rodriguez-Hertz–Ures among others.

For the proof of their ergodicity result for the WP flow, Burns, Masur and Wilkinson take part of their inspiration from the work of Katok-Strelcyn where Pesin’s theory (of existence and absolute continuity of stable and unstable manifolds) is extended to singular hyperbolic systems.

In a nutshell, the basic philosophy behind Katok-Strelcyn’s work is the following. Given a non-uniformly hyperbolic system with some non-trivial singular set, all dynamical features predicted by Pesin theory in virtue of the (non-uniform) *exponential* contraction and expansion are *not* affected *if* the loss of control on the system is at most *polynomial* as one approaches the singular set. In other terms, the hyperbolic (exponential) behavior of a singular system is not disturbed by the presence of a singular set where the first two derivatives of the system lose control in a polynomial way. In particular, this hints that Hopf’s argument can be extended to singular hyperbolic systems with polynomially bad singular sets.

In this direction, Burns-Masur-Wilkinson shows the following *ergodicity criterion* for singular hyperbolic geodesic flows (cf. Theorem 3.1 of Burns-Masur-Wilkinson’s paper).

Let be the quotient of a contractible, negatively curved, possibly incomplete, Riemannian manifold by a subgroup of isometries of acting freely and properly discontinuously.

By slightly abusing notation, we denote by the metrics on and induced by the Riemannian metric of .

We consider the metric completion of the metric space , i.e., the (complete) metric space consisting of all equivalence classes of Cauchy sequences under the relation if and only if equipped with the metric , and we define the *boundary* .

Theorem 3 (Burns-Masur-Wilkinson)Let be a manifold as above. Suppose that:

- (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).

Once Burns, Masur and Wilkinson have Theorem 3 in their toolbox, they prove the ergodicity result for WP flow in Theorem 1 by checking that the moduli space of curves (Riemann surfaces) equipped with the WP metric satisfies the six items (I) to (VI) above. Here, it is worth to point out that the most delicate items are (IV) and (VI), and Burns, Masur and Wilkinson verify them for the WP metric with the help of important previous works of McMullen and Wolpert (see these papers here).

In any case, this completes the outline of the proof of Burns-Masur-Wilkinson theorem on the ergodicity of WP flow.

**2. Rates of mixing of WP flow **

As we mentioned above, both Teichmüller and WP flows are uniformly hyperbolic in compact parts of the moduli space of curves. Since a uniformly hyperbolic system is exponentially mixing, the sole thing preventing an exponential rate of mixing for these flows is the possibility that a “big” set of orbits spends a “lot” of time near infinity (or rather the boundary of the moduli space) before coming back to the compact parts.

In the case of Teichmüller flow, the volume in Teichmüller metric of a -neighborhood of the boundary of moduli space is exponentially small (of order where denotes any fixed positive real number strictly smaller than ; cf. Corollary 2.16 of Avila-Gouëzel-Yoccoz paper).

Intuitively, this says that the “probability” that an orbit spends a long time near the boundary of moduli space is exponentially small (cf. Theorem 2.15 of Avila-Gouëzel-Yoccoz paper). In particular, the excursions near infinity of most orbits is not long enough to disrupt the exponential rate of mixing “imposed” by hyperbolic dynamics of the Teichmüller flow on compact parts. Of course, this is merely a vague intuition behind the exponential mixing of the Techmüller flow and the curious reader is encouraged to consult the articles of Avila-Gouëzel-Yoccoz and and Avila-Gouëzel for detailed explanations.

On the other hand, in the context of the WP flow, we will see that the volume in WP metric of -neighborhood of the boundary of moduli space is (see Lemma 6.1 of Burns-Masur-Wilkinson paper).

Therefore, the “probability” that an orbit of WP flow spends a long time near infinity *might* be *only* polynomially small but *not* exponentially small. In particular, this possibility might conspire against an exponential mixing of WP flow.

In fact, in our joint work with Burns, Masur and Wilkinson, we construct a subset of volume of orbits of WP flow staying near infinity for a time (at least). For this sake, we use some estimates of Wolpert (see also Propositions 4.11, 4.12 and 4.13 in Burns-Masur-Wilkinson paper) saying that the geometry of WP metric on the moduli space of curves of genus looks like a *product* of the WP metrics on the moduli spaces of curves of lower genera . In particular, the set is chosen to correspond to geodesics travelling *almost parallel* to one of the factors of the product for a relatively *long time*.

On the other hand, the existence of such sets means that the rate of mixing of WP flow can *not* be very fast.

Indeed, by taking a “smooth approximation” of the characteristic function of (i.e., supported on and ), and by letting be a *fixed* smooth function supported on the compact part (away from infinity), we see that

for . In fact, the second equality follows because is supported in the compact part of the moduli space, is supported on and the set is disjoint from the compact part for (by construction of ), so that for . Therefore, at time , we deduce that , and, hence, the correlation functions associated to WP flow can not decay faster than a polynomial function of degree of as the time .

Finally, let us remark that this argument does not work in genus because the crucial fact (in the construction of the set ) that the WP metric looks like the product of WP metrics in moduli spaces of lower genera *breaks down* in genus : indeed, in this situation, the moduli space is naturally compactified by adding a single point (because the moduli space in lower genus is trivial) and so the WP metric does not behave like a product (or, more precisely, *no* sectional curvature approaches zero as we get close to infinity). In this case, Burns, Masur, Wilkinson and myself shows that this “absence of zero curvatures at infinity” actually implies that the rate of mixing of the WP flow on the moduli space of torii is *rapid*, i.e., faster than any polynomial function of .

Concluding this section (and today’s post), let us observe that we do not know if the rate of mixing of the WP flow on moduli space of curves of genus is *genuinely* polynomial.

Indeed, recall that the naive intuition says that the rate of mixing is polynomial if we can show that most orbits do not spend long time near infinity.

Of course, this would *not* be the case if the WP metric is *very close* to a product metric, or, more precisely, if some sectional curvatures of WP metric are very close to zero: in fact, the structure of a product metric near infinity would allow for several orbits to travel almost parallel to the factors of the product (and, hence, near infinity) for a very long time.

So, we need estimates saying how *fast* the sectional curvatures of WP metric approach zero as one gets close to infinity, and, unfortunately, the best estimates available so far (due to Wolpert) do not give this type of information (because of certain *potential* cancellations in Wolpert’s calculations).

]]>

In this article, we are mainly interested in the constraints on the Lyapunov spectrum of certain linear cocycles acting in a irreducible way on the fibers.

More concretely, we consider the following setting. Let be a semisimple Lie group acting on a space . Denote by a compactly supported probability measure on and a -stationary probability on , i.e., (that is, is “invariant on -average” under push-forwards by elements of ).

A *linear cocycle* is where is a real finite-dimensional vector space. For our purposes, we will assume that the matrices are bounded for any in the support of .

From the point of view of Dynamical Systems, we think of a (linear) cocycle as the “fiber dynamics” of the following system modeling random products of the matrices while following a forward random walk on .

Let and denote by be the natural shift map where . Observe that, by -stationarity of , the product probability measure is -invariant where . For the sake of simplicity, we will assume that the -stationary measure is ergodic in the sense that is -ergodic.

In this language, the orbits of the map given by

are modeling random products of the matrices along random walks in . Indeed, this is clearly seen through the formula:

In this context, Oseledets theorem applied to ensures the existence of a collection of numbers with multiplicities called *Lyapunov exponents* and, at -almost every point , a *Lyapunov flag*

such that is a subspace of dimension and

whenever .

The collection of Lyapunov exponents and Lyapunov flags of (or, more precisely, ) is called the *Lyapunov spectrum* of .

Evidently, the Lyapunov spectrum of an “arbitrary” cocycle can exhibit “any” wild behavior. However, concerning “specific” cocycles, a general question of great interest in Dynamical Systems/Ergodic Theory is the following: what can one say about the Lyapunov spectrum of a cocycle satisfying certain geometrical and/or algebraic constraints?

Of course, this question is somewhat vague and, for this reason, there is no unique answer to it. On the other, it is precisely the vagueness of this question that makes it so appealing and this explains the vast literature providing answers for several formulations of this question.

For instance, the seminal works of Furstenberg, Goldsheid-Margulis and Guivarc’h-Raugi gave geometrical (“contraction”/“proximality” and “strong irreducibility”) and algebraic conditions (full Zariski closure of the monoid of matrices generated by the cocycle in the special linear group or in the symplectic group ) ensuring the *simplicity* of the Lyapunov spectrum (that is, the multiplicities are all equal to ). More recently, Avila-Viana gave an alternative set of geometrical conditions (“pinching” and “twisting”) ensuring simplicity for cocycles taking values in both and , and it was observed by Möller, Yoccoz and myself in this paper here that the arguments of Avila-Viana can be further extended to cocycles taking values in the classical groups , and (resp.) of real, complex and quaternionic matrices (resp.) preserving an indefinite form of signature on .

Actually, a careful inspection of these papers (in particular the ones by Golsheid-Margulis and Guivarc’h-Raugi) reveals that one can get *partial* constraints on the Lyapunov spectrum by assuming only some parts of the geometrical conditions required in these articles, and, as it turns out, this fact is already important in some applications.

More precisely, during the proof of their profound Ratner-type theorem for the -action on the moduli space of translation surfaces, Eskin-Mirzakhani needed to know a certain property of *semisimplicity* of the so-called Kontsevich-Zorich cocycle (in order to apply the “exponential drift” idea of Benoist-Quint via appropriate “time-changes”). Here, by semisimplicity we mean that, up to conjugating (or, equivalently, replacing the cocycle by for some adequate measurable map ), the cocycle is *block-conformal*, i.e.,

where ‘s are conformal (that is, belongs to an orthogonal group after multiplication by an adequate constant). Equivalently, has *semisimple Lyapunov spectrum* if we can write the quotients between consecutive subspaces of the Lyapunov flag as

where each admits a non-degenerate quadratic form such that for all and for all one has

with satisfying the cocycle relation

In this setting, Eskin-Mirzakhani needs the following fact essentially contained in the works of Goldsheid-Margulis and Guivarc’h-Raugi: a cocycle as above has semisimple Lyapunov spectrum whenever it is *strongly irreducible*, i.e., no finite cover of (or rather the induced cocycle in a finite cover of ) preserves a measurable family of proper subspaces of in the sense that for -almost every and -almost every (where is the natural measure induced in the corresponding finite cover of ).

Unfortunately, even though the experts in the subject know that this statement really follows from the ideas of Goldsheid-Margulis and Guivarc’h-Raugi, it is hard to deduce this fact *directly* from the statements in these papers.

The proof of this fact in the case of a strongly irreducible cocycle whose *algebraic hull* is is explained in Appendix C of Eskin-Mirzakhani’s paper. Here, by algebraic hull we mean the smallest -algebraic subgroup such that for some measurable (conjugation) map (by a result of Zimmer, algebraic hulls always exist and they are unique up to conjugation). Nevertheless, for their application to the Kontsevich-Zorich cocycle, they need the semisimplicity statement for the general case, i.e., without assumptions on the algebraic hull (because the Kontsevich-Zorich takes values in some other smaller classical groups in certain examples), and this is precisely one of the main purposes of our preprint with Alex Eskin.

In other terms, one of our main objectives is to adapt the ideas of Goldsheid-Margulis and Guivarc’h-Raugi to show that a strongly irreducible cocycle has semisimple Lyapunov spectrum. Also, we show in the same vein that the top Lyapunov exponent is associated to a *single* conformal block in the sense that the decomposition in (1) is trivial (i.e., ) for .

In a sense, some of the ideas of proof of the statement in the previous paragraph were previously discussed in this blog (see this post here) in some particular cases.

For this reason, we will not give here a detailed discussion of our preprint with Alex Eskin. Instead, we strongly encourage the reader that is not used to these types of arguments/ideas to replace by (or , ) in our paper with Alex Eskin and then to compare the statements there with the ones in this previous blog post here and/or in Appendix C of Eskin-Mirzakhani’s paper. By doing so, the reader will be convinced that the basic ideas are the same up to replacing some linear algebra statements by the analogous facts for the action of elements of on stationary measures supported on a flag variety (where is a certain parabolic subgroup corresponding to some subset of simple roots), etc.

Closing this post, let us just to try to summarize in a couple of words the proof of the semisimplicity of the Lyapunov spectrum of strongly irreducible cocycles . Firstly, by analyzing (in Section 2 of our preprint) the action of elements of the algebraic hull on adequate flag varieties (for some choices of subsets of simple roots), we show that can be conjugated to take its values in a certain parabolic subgroup (cf. Proposition 3.2 of our preprint). In a certain sense, this information seems of little value because, *very roughly speaking*, the fact that takes values in a parabolic subgroup essentially amounts to say that we can put in the form

and this is certainly not the desired block conformality property (as this last property means that all entries above are all zero). Nonetheless, we combine the information takes values in a certain parabolic subgroup with the *analogous* statement for the *backward cocycle* (i.e., the cocycle obtained by following the bacwards random walk on ) to deduce that is *Schmidt-bounded*, i.e., it is uniformly bounded on large compact sets of almost full measure. By a result of Schmidt (discussed in this previous blog post), up to conjugation, any Schmidt-bounded cocycle takes values in a compact subgroup, and from this last fact one can establish that is block-conformal (i.e., all ‘s are zero in the equation above). Finally, the statement that the top Lyapunov exponent corresponds to a single conformal block essentially follows from the well-known fact that the highest weight of the *irreducible* action of the algebraic hull of the strongly irreducible cocycle on the real finite-dimensional vector space has multiplicity .

]]>

This workshop is mainly organized by Gugu Moreira (and Yuri Lima, Christian Mauduit, Jean-Christophe Yoccoz and myself served as co-organizers).

The schedule of the workshop is available at the link provided above, and, as the reader can check, there will be several exciting talks by several experts on these subjects (e.g., Vitaly Bergelson, Alex Eskin, Harald Helfgott, Elon Lindenstrauss, Janos Pintz, etc.).

Closing this short post, let me point out that the videos of the lectures are available here (and they are uploaded automatically at IMPA’s website by the end of each lecture), and, in fact, it is even possible to watch online in real time the lectures at this link here (see the sublink next to the name “Ricardo Mañé”).

]]>

In their (long) proof of this result, A. Eskin and M. Mirzakhani use several arguments inspired by the *low entropy method* of M. Einsiedler, A. Katok and E. Lindenstrauss, the *exponential drift argument* of Y. Benoist and J.-F. Quint and, as a preparatory step for the exponential drift argument, they show the *semisimplicity* of the Kontsevich-Zorich cocycle.

In Eskin-Mirzakhani’s article, the proof of the semisimplicity property of the Kontsevich-Zorich cocycle is based on the work of G. Forni and the study of *symplectic* and *isotropic* -invariant subbundles of the Hodge bundle.

It is interesting to point out that, while *symplectic* -invariant subbundles of the Hodge bundle occur in several known examples (see, e.g., these articles here), the existence of some example of *isotropic* -invariant subbundle is not so clear.

Indeed, the question of the existence of non-trivial isotropic -invariant subbundles of the Hodge bundle was posed by Alex Eskin and Giovanni Forni (independently) and they were partly motivated by the fact that the non-existence of such subbundles would allow to “forget” about isotropic -invariant subbundles and thus, simplify (at least a little bit) some arguments in Eskin-Mirzakhani paper.

In this note here, Gabriela Schmithüsen and I answered this question of A. Eskin and G. Forni by exhibiting a square-tiled surface of genus with squares such that the Hodge bundle over the -orbit of has non-trivial isotropic -invariant subbundles.

Fortunately, the basic idea of this example is simple enough to fit into a (short) blog post, and, for this reason, we will spend the rest of this post explaining the general lines of the construction of (leaving a few details to the our note with Gabi).

**1. Preliminaries **

For the sake of convenience of the reader, we reproduce below Section 2 of our note with Gabi where some key facts about translation surfaces are recalled. Of course, this section is far from being an appropriate introduction to this subject and the reader might want to consult A. Zorich’s survey, and/or these posts here for a gentle expositions on the moduli spaces of Abelian differentials. Also, the reader may find useful to consult the introduction of our article with J.-C. Yoccoz for further comments on the relationship between the Kontsevich-Zorich cocycle and the action on homology of affine diffeomorphisms of translation surfaces.

A *translation surface* is the data of a non-trivial Abelian differential on a Riemann surface . This nomenclature comes from the fact that the local primitives of outside the set of its zeroes provides an atlas on whose changes of coordinates are all translations of the plane . In the literature, these charts are called *translation charts* and an atlas formed by translation charts is called *translation atlas* or *translation (surface) structure*. For later use, we define the *area* of as .

The *Teichmüller* space of *unit area* Abelian differentials of genus is the set of unit area translation surfaces of genus modulo the natural action of the group of orientation-preserving homeomorphisms of isotopic to the identity. The *moduli* space of *unit area* Abelian differentials of genus is the set of unit area translation surfaces of genus modulo the natural action of the group of orientation-preserving homeomorphisms of . In particular, where is the *mapping class group* (of isotopy classes of orientation-preserving homeomorphisms of ).

The point of view of translation structures is useful because it makes clear that acts on the set of Abelian differentials : indeed, given , we define as the translation surface whose translation charts are given by post-composing the translation charts of with . This action of descends to and . The action of the diagonal subgroup of is the so-called *Teichmüller (geodesic) flow*.

Remark 1By collecting together unit area Abelian differentials with orders of zeroes prescribed by a list of positive integers with , we obtain a subset of calledstratumin the literature. From the definition of the -action on , it is not hard to check that the strata are -invariant.

The *Hodge bundle* over is the quotient of the trivial bundle by the natural action of the mapping-class group on *both* factors. In this language, the *Kontsevich-Zorich cocycle* is the quotient of the trivial cocycle

by the mapping-class group . In the sequel, we will call as KZ cocycle for short.

For our current purposes, let us restrict ourselves to the class of translation surfaces covering the square flat torus equipped with the Abelian differential induced by on . In the literature, these translation surfaces are called *square-tiled surfaces* or *origamis* because is tiled by the (open) squares given by the pre-image of the open unit square . In particular, by labeling these open squares , we see that a square-tiled surface determines a pair of permutations coding the successive appearances of squares along the horizontal and vertical directions, and *vice-versa*.

The stabilizer — also known as *Veech group* — of a square-tiled surface with respect to the action of is commensurable to , and its -orbit is a *closed* subset of isomorphic to the unit cotangent bundle of the hyperbolic surface .

The Veech group consists of the “derivatives” (linear parts) of all *affine diffeomorphisms* of , that is, the orientation-preserving homeomorphisms of fixing the set of zeroes of whose local expressions in the translation charts of are affine maps of plane. The group of affine diffeomorphisms of is denoted by and it is possible to show that is precisely the subgroup of elements of stabilizing in . The Veech group and the affine diffeomorphisms group are part of the following exact sequence

where, by definition, is the subgroup of *automorphisms* of , i.e., the subgroup of elements of whose linear part is trivial (i.e., identity).

In this language, the KZ cocycle on the Hodge bundle over the -orbit of is intimately related to the action on homology of . Indeed, since is the stabilizer of in , we have that the KZ cocycle is the quotient of the trivial cocycle

by .

For later use, we observe that, given a square-tiled surface (where is a finite cover ramified precisely over ), the KZ cocycle, or equivalently , preserves the decomposition

where and .

Closing this preliminary section, we recall that, given a finite ramified covering of Riemann surfaces, the *ramification data* of a point is the list of ramification indices of all pre-images of counted with multiplicities.

**2. Forni’s subbundle **

Before trying to construct examples of isotropic -invariant subbundles of the Hodge bundle, we need to have a clue where they can possibly be found. Here, we are in good shape because, by Theorem A.6 and A.4 in Appendix A of Eskin-Mirzakhani’s paper, we have:

Theorem 1Let be an isotropic -invariant subbundle of the Hodge bundle. Then, all Lyapunov exponents of the restriction of the Kontsevich-Zorich cocycle to vanish and, furthermore, the Kontsevich-Zorich cocycle acts isometrically on with respect to an adequate (Hodge) norm.

In other words, this theorem says that all isotropic -invariant subbundles of the Hodge bundle must live inside the maximal -invariant subbundle where the Kontsevich-Zorich cocycle acts isometrically. In the literature, the subbundle is called *Forni’s subbundle* and some of its properties were studied in these two articles here.

In particular, it is worth to look first at examples where Forni’s subbundle is not trivial before searching for isotropic -invariant subbundles. As it turns out, such examples are known: the Eierlegende Wollmichsau origami, the Ornithorynque origami and, more generally, certain square-tiled cyclic covers are some examples where the Forni subbundle is not trivial.

However, by a closer inspection of these examples, one can check that, even though Forni’s subbundle is not trivial in these examples, there are no isotropic -invariant subbundles simply because there are no *proper* -invariant subbundles (inside Forni’s subbundle of these examples) at all! For instance, the fact that this happens for the Eierlegende Wollmilchsau and Ornithorynque was shown in this article here.

In other words, despite the non-triviality of Forni’s subbundle in these examples, the fact that Forni’s subbundle is -irreducible makes that we don’t have a chance to find isotropic -invariant subbundles for the Eierlegende Wollmilchsau (say).

**3. Isotropic -invariant subbundles **

We just saw that the Eierlegende Wollmichsau and some other square-tiled cyclic covers are not the examples we are looking because the Kontsevich-Zorich cocycle acts irreducibly on their (non-trivial) Forni subbundles. On the other hand, the results in this article here indicate that these examples are not very far from being the desired ones. Indeed, it is shown in this article that the Kontsevich-Zorich cocycle acts on the Forni subbundle of the Eierlegende Wollmilchsau via a *finite group* (computed explicitly in the paper). In particular, by taking adequate finite covers of , the Kontsevich-Zorich cocycle will act trivially on the corresponding piece of Forni subbundle and thus one can eventually get rid the irreducibility issue.

This idea is the key tool in our note with Gabi and, as it turns out, it is sufficiently simple so that one can even construct explicit examples with isotropic -invariant subbundles of the Hodge bundle.

More precisely, we start with the Eierlegende Wollmilchsau . The Forni subbundle in this case is a -dimensional symplectic subbundle of the Hodge bundle over the orbit of and the elements of the affine group (or equivalently the Kontsevich-Zorich cocycle) with linear part (derivative) in the congruence subgroup

and fixing the zeroes of act trivially on . See, e.g., our article with J.-C. Yoccoz for a proof of these facts.

Therefore, *if* we can construct a finite cover such that all affine diffeomorphisms of “descend” to an affine diffeomorphisms of (in the sense that ) with linear part in and fixing the zeroes of , *then* is our desired example. Indeed, in this setting, it follows that the action of any affine diffeomorphism of on the -dimensional subbundle occurs through the action on of some affine diffeomorphism of with linear part in . On the other hand, as we mentioned above, the action of any such is trivial. So, we deduce that the action of any affine diffeomorphism of on the -dimensional subbundle is trivial. In particular, *any* line (-dimensional subspace) of defines an isotropic subbundle invariant under the whole affine group, i.e., any line of induces an isotropic -invariant subbundle.

Finally, it remains only to know what are some conditions so that a finite cover has the property that all affine diffeomorphisms “descend” to affine diffeomorphisms with derivative in fixing the zeroes of . Intuitively, is composed of several copies of glued in some way. Thus, what could go “wrong” when trying to “descend” an affine diffeomorphism is that the equation might not define a meaningful object because mixes up the several copies of in some strange way. In order to avoid this problem, the idea is to construct so that the ramification data over certain special points are different: this makes that these points are geometrically different from each other and so they can’t be mixed together by affine diffeomorphisms; in particular, these points prevent the copies of containing them to mix up in strange ways. More concretely, in Proposition 2 of our note with Gabi, we show that it suffices prescribe different ramification data for the covering over *some* -torsion. By doing so in a somewhat “minimalist” way (such as in Lemma 3.1 of our note with Gabi), one ends up with the following concrete example of square-tiled surface of genus 15 and 512 squares with isotropic -invariant subbundles:

]]>

More precisely, using the notations of this post (as well as of its companion), we mentioned that a lattice of can be equipped with a probability measure such that the Poisson boundary of coincides with the Poisson boundary of equipped with any spherical measure (cf. Theorem 13 of this post). Then, we sketched the construction of the probability measure in the case of a cocompact lattice of , and, after that, we outlined the proof that is a boundary of in the cases and .

However, we skipped a proof of the fact that is the Poisson boundary of by postponing it possibly to another post. Today our plan is to come back to this point by showing that is the Poisson boundary of .

More concretely, we will show the following statement due to Furstenberg. Let be a cocompact lattice of . As we saw in this previous post (cf. Proposition 14), one can construct a probability measure on such that

- (a)
*has full support*: for all , - (b)
*is*-*stationary*: , - (c)
*the*-*norm function is*-*integrable*: .

Here, we recall (for the sake of convenience of the reader) that: is the “complete flag variety” of or, equivalently, where is the subgroup of upper-triangular matrices, is the Lebesgue (probability) measure and

where acts on Poincaré’s disk via Möebius transformations (as usual) and denotes the hyperbolic distance on Poincaré’s disk .

Then, the result of Furstenberg that we want to show today is:

Theorem 1Let be a cocompact lattice of and denote by any probability measure on satisfying the conditions in items (a), (b) and (c) above. Then, the Poisson boundary of is .

The proof of this theorem will occupy the entire post, and, in what follows, we will assume familiarity with the contents of these posts.

**1. Preliminaries **

As we already mentioned above, we know that is a boundary of (cf. Subsection 2.2 of this post).

Thus, if we denote by the Poisson boundary of (an object constructed in Section 4 of this post), then, by the maximality of the Poisson boundary, is an *equivariant image* of under some *equivariant* map .

Our goal consists into showing that is an *isomorphism*, and, for this sake, it suffices to show that we can recover all bounded measurable functions of from the corresponding functions on via , i.e., the proof of Theorem 1 is reduced to prove that:

In this direction, it is technically helpful to replace by and consider the subspace

In fact, since is a *closed* subspace of the *Hilbert space* , we have an orthogonal projection and our task of proving Proposition 2 is equivalent to show that is the identity map .

Now, the basic strategy to show that is to prove that, for each , the functions and induce the *same* -harmonic function on (via *Poisson formula*). Indeed, since is the Poisson boundary of , we have (by definition) that the Poisson formula associates an *unique* -harmonic function

on to each . Hence, if and are associated to the same -harmonic function on , then . In other words, we reduced the proof of Proposition 2 to the following statement:

Proposition 3Given , the functions and induce the same -harmonic function on via Poisson formula.

In other to show this proposition, we rewrite the -harmonic function associated to in terms of the -inner product as follows:

In particular, *if* for *all* , then

Equivalently, we just showed that and induce the same -harmonic function *if* for all , that is, the proof of Proposition 3 will be complete once we prove that:

As it turns out, the functions admit a nice characterization in terms of Jensen’s inequality. More concretely, since consists of all functions in which are measurable with respect to the field of sets (with measurable), one can show that the projection enjoys a “Jensen’s inequality property”:

with equality holding *only* for functions .

As the reader might suspect, we intend to use Jensen’s inequality to produce an equality characterizing whether . For this, we will compute for .

In fact, it is not hard to *guess* who must be: since is an equivariant map sending to , it is not surprising that . Now, let us formalize this naive guess as follows. Recall that, by definition, is the (unique) function in such that

for each , i.e., with ). We rewrite this identity as

For , this identity becomes

Observe that the right-hand side of this equality is the -harmonic function of induced by . On the other hand, since is an equivariant map between the Poisson boundary and the boundary , we have that the functions and induce the same -harmonic function, i.e.,

By putting the previous two equalities together, we get that

Next, we recall that sends to (i.e., ). Therefore, if we denote , we obtain that the right-hand side of the previous equality becomes

By combining the last two equalities above, we deduce that

Since this identity holds for an arbitrary function , we conclude that

as it was claimed (or rather guessed).

From this computation and Jensen’s inequality (1), we get the following lemma:

*Proof:* By setting , we see that the left-hand side of (2) is

while our computation of above reveals that the right-hand side of (2) is

It follows that the desired lemma is a consequence of Jensen’s inequality (1).

This lemma reduces to proof of Proposition 4 to show that one has an equality in (2) (for all ). Here, we claim that it is sufficient to check that

for all . Indeed, since has full support, i.e., for all (cf. item (a) above), it follows from (2) and (3) that one has equality in (2) for all .

In summary, our task now becomes to prove that:

Proposition 6For all , the inequality (3) above holds, i.e.,

The basic idea to prove this proposition is the following. The quantities and can be interpreted as spatial averages. In particular, the ergodic theorem will tell us that and drive the Birkhoff sums of the observables and along almost every sample of random walk in .

Now, assuming by contradiction that , we will see that the Birkhoff sums of are *very well controlled* by the Birkhoff sums of (with some “margin” coming from the strict inequality ). Using this and the fact that the density can be explicitly computed, we will be able to solve a *counting problem* to show that:

Proposition 7If , then there exists arecurrence subsetof (i.e., a subset that is hit by the random walk infinitely often with probability ) with the property that

On the other hand, using the properties of -harmonic functions, we will show the following general fact about recurrence sets of :

Proposition 8Let be a recurrence set of for the random walk associated to a stationary sequence of independent random variables with distribution . Then,

Of course, by putting together Propositions 7 and 8, we deduce the validity of Proposition 6. Hence, it remains only to prove Propositions 7 and 8. In order to organize the discussion, we will show them in separate sections, namely, the next section will concern Proposition 7 while the final section of this post will concern Proposition 8.

**2. Proof of Proposition 7 **

As we already mentioned above, the first step in the proof of this proposition is to observe that and are *spatial* averages, so that the ergodic theorem says that one can express them in terms of *temporal* averages along typical “orbits” (samples of random walk).

More precisely, let be a stationary sequence of independent random variables with distribution and consider the -process on . For *technical* reasons (that will become clear in a moment), we will think of as moving *forward* in time (rather than backward), i.e., the -process satisfies

with independent of (instead of and independent of ). Note that by setting

we get a -process on (because is an equivariant map from to ).

** 2.1. Interpretation of as a Birkhoff sum **

In this language, we can convert the spatial average of the observable

in a Birkhoff average as follows. Let us consider the random walk on obtained by *left*-multiplication. Then,

By applying the ergodic theorem to the right-hand side of this expression (and using the fact that and are independent), we obtain that

Of course, in order to justify the application of the ergodic theorem, we need to check the (absolute) integrability of the corresponding observable, that is, we need to show that the following expectation

is finite.

As it turns out, the finiteness of this expectation is a consequence of the integrability condition on in item (c) above. Indeed, we have

can be controlled as follows. By letting act on the Poincaré’s disk via

where and . We have that

A simple calculation using this expression and the fact that reveals that

Therefore, from the -integrability of , cf. item (c) above, we deduce that

and, in view of (6), we conclude the integrability of (5).

In summary, the validity of (4) essentially follows from the -integrability condition on in item (c).

** 2.2. Interpretation of as a Birkhoff sum **

Similarly to the case of , we want to convert into Birkhoff average. Again, let us consider the random walk on , and let us write

We want to apply once more the ergodic theorem to obtain

However, the justification of the application of the ergodic theorem is a little bit more *subtle* because the (absolute) integrability of

might be *not* true. Indeed, we have *no* prior information on the relationship between and , so that we can not use item (c) to get the integrability (contrary to the case of the *Lebesgue* measure where could be computed *explicitly*). Fortunately, it is not hard to overcome this little technical difficulty: as it turns out, the ergodic theorem also applies to observables that are *bounded only on one side* by a -integrable function; in particular, we can apply the ergodic theorem to because

** 2.3. Construction of a “weird” recurrence set when **

During this subsection, let us *assume* that . Recall that the plan is to show that the Birkhoff sums of of are *very well controlled* by the Birkhoff sums of .

In this direction, let us observe that the asymptotics in (4) and (7) imply

Using the properties of the Radon-Nikodym derivative (e.g., ), we can rewrite the numerator in the left-hand side of this equation as:

From this and (9) we deduce that

with probability . Since ‘s are independent of , we conclude from (11) that, for almost every , one has

for almost all random paths .

In particular, we can fix two *distinct* values and of so that (11) holds for almost every random path. For , let us consider the random variables

and

We are interested in the properties of (as is the random walk on ) but (11) provides information only about . Fortunately, and have the *same* distribution, so that all probabilistic statements about are also true for . In particular, for each , the probabilities of the events

go to for because the probabilities of the events

go to for in view of the fact that (11) implies

with probability (for ).

Therefore, if we choose a sequence going *very fast* to infinity as so that the *sum* of the probabilities of the events

is finite (for ), then we can use the Borel-Cantelli lemma to obtain that

with probability . In particular, it follows that the set

is a *recurrence* set for the random walk (i.e., this random walk visits infinitely often with probability ).

Now, if , we can take and such that

In other words, the density is very well-controlled by with a “margin” coming from the assumption that .

From this nice control our plan is to prove that the recurrence set has the “weird” property referred to in Proposition 7, i.e., we will show that

Keeping this goal in mind, given , let us denote by

By (12), we can bound the quantity as follows:

where

and we observe that , we can estimate the right-hand side of (13) as

Since was chosen so that (assuming ), we have that the right-hand side of this estimate is convergent *if* we can show that grows *linearly* (at most), i.e., the proof of Proposition 7 is complete once we can handle the *counting problem* of showing that

We will exploit the explicit nature of the densities in order to show this (counting) lemma. More precisely, given , recall that

if acts on Poincaré’s disk as with and .

Since and are *distinct*, the complex number can’t be close to both of them at the same time. Using this information, the reader can see that

for some constants and . Equivalently, since , one has

for some constants and .

In particular, Lemma 9 is equivalent to show that where

Actually, since the subset of elements with is *finite* (namely, it is the intersection of the lattice with the compact subgroup stabilizing ), we can convert the counting problem

for elements of into the following *geometrical* counting problem about *points* :

where

Now, this geometrical counting problem is not hard to solve, at least when is cocompact.

Indeed, let us consider first a *large* compact subset of containing a fundamental domain of about the origin . Then, by definition, the -translates of cover and, hence,

where is an appropriate constant (depending on ) and is the area of the hyperbolic disk of radius centered at .

Next, let us consider a *small* compact ball of around so that it is disjoint from its -translates. Then, we have that

where is an appropriate constant (depending on ).

In summary, there are two constants and such that

On the other hand, the area of the hyperbolic disk of radius centered at is not hard to compute:

where is the Euclidean radius of , i.e., . From this expression we see that

so that this ends the proof of Lemma 9.

This completes the proof of Proposition 7.

**3. Proof of Proposition 8 **

Closing this post, let us show that the properties of -harmonic functions do not allow the existence of the “weird” recurrence sets constructed in Proposition 7. For this sake, let us suppose by *contradiction* that is a recurrence subset of such that

By removing *finitely* many elements of if necessary, we get a recurrence set that we still denote such that

Next, let us observe the following facts. Firstly, since is -stationary and is fully supported on (cf. item (a) above), we have that is absolutely continuous with respect to and the density is bounded because

so that . Secondly, from the previous identity, we see that

so that, for almost every , the function

is -harmonic.

In particular, our plan is to use the mean value property of -harmonic functions to express the values of in terms of its values in in order to eventually contradict (14).

For this sake, let us show the following elementary abstract lemma about the mean value property of *bounded* -harmonic functions with respect to recurrence sets:

Lemma 10Let be a discrete group with a probability measure and denote by a stationary sequence of independent random variables with distribution . If is a recurrence set of the random walk and is a bounded -harmonic function on , then the following mean value property with respect to holds:

where is the distribution of the first point of hit by .

*Proof:* We start with the usual mean value property

Now, for each term we can *independently* decide whether we want to use again the mean value relation to express as a convex combination of or not. Since our ultimate goal is to write as a convex combination of the values of on the recurrence set , we will take our decision as follows: if , we leave alone, and, otherwise, we apply the mean value relation.

After steps of this procedure, we have

where *“something”* is a combined weight of contributions coming from the values of on points *outside* that were reached by the random walk after steps.

Because is a recurrence set, the random walk reaches with probability . Therefore, since the function is bounded, we can pass to the limit as in the identity above to get the desired equality

This proves the lemma.

Coming back to the context of Proposition 8, we observe that this lemma does *not* apply directly to the -harmonic density function

because it might be *unbounded*.

Nevertheless, by revisiting the argument of the proof of the lemma above, one can easily check that, for an unbounded -harmonic (integrable) function , one has the mean value *inequality*

(but possibly *not* the mean value equality ) where is the probability that the first point of hit by the random walk is .

In any event, using this mean value inequality with we deduce that

for almost every .

In particular, we conclude that

Thus, in view of (14), we obtain that

that is, the total probability that the random walk hits is *strictly* smaller than , a contradiction with the fact that is a recurrence set of the random walk!

This completes the proof of Proposition 8, and, hence, this finishes the sketch of proof of Furstenberg’s Theorem 1.

]]>