Hello! This week I will be teaching the second-half of a minicourse (together with Giovanni Forni) entitled “*Introduction to Teichmuller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards*”. This minicourse is part of the activities of the School and Workshop “Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory” held at Banach Center, Bedlewo, Poland. The conference schedule is very intense (we have minicourses from 9AM to 6PM [and seminars from 8Pm to 9PM on Tuesdays and Thursdays]) and we had several advanced minicourses by Federico Rodriguez-Hertz, Manfred Einsiedler, Anatole Katok and Giovanni Forni on topics running from higher-rank hyperbolic and partially hyperbolic actions, measure rigidity, Ratner theory and Teichmüller dynamics theory. For next week, there will be minicourses by Mike Hochman, Livio Flaminio, Zhenqi Wang (and myself). The minicourses by Z. Wang and myself are continuations of the minicourses by A. Katok and G. Forni respectively, while the minicourses by M. Hochman and L. Flaminio concerns Ergodic Theory and Fractal Geometry, and Applications of Representation Theory to Dynamics (resp.).

In any case, this minicourse was an excellent opportunity to complete the fourth post of my series on Teichmüller dynamics. Below the fold, you’ll find (as promised a few months ago) a discussion of the Ergodic Theory of the Teichmüller flow with respect to Masur-Veech measures and rough sketches of its applications to interval exchange transformations and translation flows.

**1. Ergodic theory of Teichmüller flow with respect to Masur-Veech measure **

Let be a connected component of a stratum of Abelian differentials with unit area, and denote by the corresponding Masur-Veech probability measure.

** 1.1. Finiteness of Masur-Veech measure and its applications to interval exchange maps **

One important application of the fact that the Teichmüller flow preserves a natural (Masur-Veech) probability measure is the *unique ergodicity* of “almost every” *interval exchange transformation* (i.e.t. for short). Recall that an *i.e.t.* is a map where are subsets of an open bounded interval such that and are finite sets and the restriction of to each connected component of is a translation onto some connected component of . For some concrete examples, see Figure 1 below.

It is not hard to see that an i.e.t. is determined by a *metric data*, i.e., lengths of the connected components of , and *combinatorial data*, i.e., a permutation indicating how the connected components of are “rearranged” after applying to them. For instance, in the example of Figure 1 where 4 intervals are exchanged, the combinatorial data is the permutation with .

In particular, it makes sense to talk about “almost every” i.e.t.: it means that a certain property holds for almost every choice of metric data with respect to the Lebesgue measure.

Remark 1In the sequel, we willalwaysassume that the combinatorial data isirreducible, i.e., if is a permutation of elements , we require that, for every , . The meaning of this condition is very simple: if is not irreducible, there is such that and hence we can studyanyi.e.t. with combinatorial data by juxtaposing two i.e.t.’s (one with intervals and another with intervals).

By applying their result on the finiteness of the mass of (see Theorem 2 of the previous post of this series), H. Masur and W. Veech deduced that:

Theorem 1 (H. Masur, W. Veech)Almost every i.e.t. is uniquely ergodic.

Philosophically speaking, the derivation of this result from the finiteness of the mass of is part of a long tradition (in Dynamical Systems) of “plough in parameter space, and harvest in phase space” (as it was said by Adrien Douady about complex quadratic polynomials and Mandelbrot set). In broad terms, the idea is that given a paramter family of dynamical systems and an appropriate renormalization procedure (defined at least for a significant part of the parameters), one can often infer properties of the dynamical system for “typical parameters” by studying the dynamics of the renormalization.

For the case at hand, we can describe this idea in a nutshell as follows. An i.e.t. can be “*suspended*” (in several ways) to a translation surface by means of the so-called Veech’s zippered rectangles construction. For example, in the Figure 2 below (extracted from A. Zorich’s survey), we see a genus 2 surface (obtained by glueing the opposites sides of the polygon marked with the same name by translation) presented as a suspension of an i.e.t. with combinatorial data . To see that this is the combinatorial data of the i.e.t., it suffices to “compute” the return map of vertical translation flow to the special segment in the “middle” of the polygon.

By definition, is the first return time map of the *vertical translation flow* of the Abelian differential to an appropriate horizontal separatrix associated to some singularity of . Here, the vertical translation flow associated to a translation surface is the flow obtained by following (with unit speed) vertical geodesics of the flat metric corresponding to . In particular, since the flat metric has singularities (in general), is defined *almost* everywhere (as vertical trajectories are “forced” to stop when they hit singular points [zeroes] of )! See the Figure 3 below for an illustration of these objects. There one can see an orbit through a point () hitting a singularity in finite time (and hence stopping by then) and an orbit through a point () whose orbit never hits a singularity (and hence it can be prolonged forever).

In particular, we can study orbits of by looking at orbits of the vertical flow on . Here, the idea is that long orbits of the vertical flow can wrap around a lot on , so that a natural procedure is to use Teichmüller flow to make the long vertical orbit shorter and shorter (so that it wraps less and less), thus making it reasonably easier to analyse. I.e., one uses Teichmüller flow to *renormalize* the dynamics of the vertical flow on translation surfaces (and/or i.e.t.’s). Of course, the price we pay is that this procedure *changes* the shape of (into ). But, if the Teichmüller flow has nice *recurrence* properties (so that the shape is very close to for appropriate choice of large ), one can hope to bypass the difficulty imposed by the change of shape.

In the case of showing unique ergodicity of almost every i.e.t., H. Masur and W. Veech observed that this can be derived from Poincaré’s recurrence theorem applied to Teichmüller flow endowed with Masur-Veech measure. Of course, for this application of Poincaré recurrence theorem, it is utterly important to know that Masur-Veech measure is a probability (i.e., it has finite mass), a fact ensured by Theorem 2 of the previous post of this series.

Evidently, this is a very rough sketch of the proof of Theorem 1. For more details, see J.-C. Yoccoz survey for a complete proof using Rauzy-Veech induction. We may come back to this point later in future posts.

Notice that the same kind of reasoning as above indicates that the unique ergodicity property must also be true for “almost every” translation flow in the sense that the vertical translation flow on almost every translation surface structure is uniquely ergodic. Indeed, the following theorem (again by H. Masur and W. Veech) says that this is the case:

Theorem 2 (H. Masur, W. Veech)Almost every translation flow is uniquely ergodic.

In the sequel, we will present a sketch of proof of this result (based on the recurrence of Teichmüller flow) assuming that the simplicity of the top exponent (a fact that I discussed sometime ago in these notes [in a ``pre-blog'' era]). We start by assuming that the vertical translation flow of our translation surface is *minimal*, that is, every orbit defined for all times are dense: this condition is well-known to be related to the absence of saddle connections (see, e.g., J.-C. Yoccoz survey), and the last property has full measure (since the presence of saddle connections for corresponds to a countable set of directions , and the Masur-Veech measure is natural).

Now, given an ergodic -invariant probability , consider a -typical point, and . Let be the homology class obtained by “closing” the piece of (vertical) trajectory with a bounded (usually *small*) segment connecting to . A well-known theorem of S. Schwartzman says that

In the literature, is called *Schwartzman asymptotic cycle*. By Poincaré duality, the Poincaré dual of gives us a class . Geometrically, is related to the *flux* of through transverse closed curves with respect to . More precisely, given a closed curve transverse to , the flux is

For the sake of the subsequent discussion, we recall that any -invariant probability induces a *transverse measure* on pieces of segments transverse to : indeed, we define by the flux through , i.e., . Since is simply a translation along the leaves of the vertical foliation of , we see that can be locally written as in any “product” open set of the form not meeting singularities of (where is a transverse segment).

We claim that the map is *injective*. Indeed, given two ergodic -invariant probabilities and with , we observe that the *transverse measures* and induced by them on a closed curve transverse to differ by the derivative of a continuous function on . Indeed, can be obtained by integration: by fixing an “origin” and an orientation on , we declare and , where is the segment of going from to in the positive sense (with respect to the fixed orientation). Of course, the fact that is well-defined, i.e., it produces the same value for when we go around is guaranteed by the assumption . Now, we note that is invariant under the return map induced by , so that, by minimality of , we conclude that the continuous function must be constant. Therefore, , i.e., and have the same transverse measures. Since and are the Lebesgue measure along the flow direction, we obtain that , so that the claim is proved.

Next, we affirm that (or equivalently ) decays exponentially fast *like* under KZ cocycle whenever the Teichmüller flow orbit of is *recurrent*. Indeed, let us fix such that is very close to , and we consider the action of KZ cocycle on . Since, by definition, is approximated by as , we have that

On the other hand, since contracts the vertical direction by a factor of and is essentially a vertical trajectory (except for a *bounded* piece of segment connecting to ), we get

where is the *stable norm* on with respect to (obtained by measuring the length of [primitive] closed curves [i.e., elements of ] using the flat structure induced by and extending this “by linearity”). In the previous calculation, we implicitly used the fact that is *very close* to , so that the stable norms and are *comparable by definite factors*, and thus the factor of can “*kill*” eventual (*bounded*) error terms coming from the “closing” procedure used to define . Therefore, our affirmation is proved.

Finally, we note that the assumption (i.e., simplicity of the top KZ cocycle exponent) means that there is only *one* direction in which is contracted like ! (namely, ) Therefore, given with minimal vertical translation flow and recurrent Teichmüller flow orbit, any -invariant ergodic probability satisfies . Since preserves the Lebesgue measure (flat area induced by ), we obtain that any -invariant ergodic probability is a multiple of , and, *a fortiori*, . Thus, is uniquely ergodic for such ‘s. Since we already saw that almost every has minimal vertical translation flow, we have only to show that -almost every is recurrent under Teichmüller flow to complete the proof of Theorem 2, but this is immediate from Poincaré’s recurrence theorem (since Teichmüller flow preserves the Masur-Veech measure , a *finite mass* measure).

** 1.2. Ergodicity of Teichmüller flow **

In the fundamental papers here and here, H. Masur and W. Veech independently showed the following result:

Theorem 3 (H. Masur, W. Veech)The Teichmüller geodesic flow is ergodic, and actually mixing, with respect to .

For a complete proof of this result using Rauzy-Veech induction, see (again) J.-C. Yoccoz survey (we may come back to this point in a future post).

Concerning the first part of the statement, we observe that the ergodicity of the Teichmüller flow is essentially a consequence of the simplicity of the top exponent and the existence of nice (“long”) stable and unstable manifolds for . Indeed, as we already know, the simplicity of the top exponent implies that, except for the zero Lyapunov exponent coming from the flow direction, the Teichmüller flow has no other zero exponent (since is the second smallest non-negative exponent). In other words, the Teichmüller flow is *non-uniformly hyperbolic* in the sense of the Pesin theory. This indicates that *Hopf’s argument* may apply in our context. Recall that Hopf’s argument starts by observing that ergodic averages are constant along stable and unstable manifolds: more precisely, given a point such that the ergodic average

exists for a (uniformly) continuous observable , then the ergodic averages

exists and for any in the stable manifold of . Actually, since , we have , so that, by the uniform continuity of , the desired claim follows. Of course, a similar result for ergodic averages along unstable manifolds holds if we replace by in the definition of . Now, the fact that we consider “future” () ergodic averages along stable manifolds and “past” () ergodic averages along unstable manifolds is not a major problem since Birkhoff’s ergodic theorem ensures that these two “types” of ergodic averages coincide at almost every point.

In particular, since the ergodicity of is equivalent to the fact that is constant at almost every point, if one could *access* any point starting from any point using pieces of stable and unstable manifolds like in Figure 4 below, we would be in good shape (here, we’re skipping some details because Hopf’s argument needs that the intersection points appearing in Figure 4 to satisfy Birkhoff’s ergodic theorem; in general, this is issue is strongly related to the so-called *absolute continuity property* of the stable and unstable manifolds, but this is not a problem in our context since Pesin’s theory ensures absolute continuity of and ).

However, it is a general fact that Pesin theory of non-uniformly hyperbolic systems only provides the existence of *short* stable and unstable manifolds. Even worse, the function associating to a typical point the size of its stable/unstable manifolds is only *measurable*. In particular, the nice scenario drew below may not happen in general (and actually the best Hopf’s argument [alone] can do is to ensure the presence of a countable number of ergodic components [at most]).

Fortunately, in the specific case of Teichmüller flow, one can determine *explicitly* the stable and unstable manifolds: since acts on by multiplying by and by , we infer that

and

In particular, we see that these invariant manifolds are “large” subsets corresponding to affine subspaces in period coordinates. Therefore, the potential problem pointed out in the previous paragraph doesn’t exist, and one can proceed with Hopf’s argument to eventually derive the ergodicity of Teichmüller flow with respect to Masur-Veech measure .

Concerning the second part of the statement of this theorem, we should say that the mixing property of Teichmüller flow is a consequence of its ergodicity and the mere *existence* of the -action: indeed, while ergodicity alone doesn’t imply mixing in general (e.g., irrational rotations of the circle are ergodic but not mixing), the fact that Teichmüller flow is part of a whole -action permits to derive mixing from ergodicity in view of the nice *representation theory* of . We will come back to this point later in this post when we discuss *exponential mixing* property of Teichmüller flow.

** 1.3. Kontsevich-Zorich conjecture (after G. Forni, and A. Avila M. Viana) **

Around 1996, A. Zorich and M. Kontsevich performed several numerical experiments leading them to conjecture that the Lyapunov spectra of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures are *simple*, i.e., the multiplicity of each Lyapunov exponent , is 1:

As we discussed in the previous post of this series, the Kontsevich-Zorich cocycle is *symplectic*, so that its Lyapunov exponents (with respect to *any*invariant ergodic probability ) are symmetric with respect to the origin: . Also, the top Lyapunov exponent is always simple (i.e., ). Therefore, the Kontsevich-Zorich conjecture is equivalent to

In 2002, G. Forni was able to show that via variational formulas (inspired by M. Kontsevich’s work) for the so-called *Hodge norm* and certain formulas for the sum of the Lyapunov exponents of the KZ cocycle (inspired by M. Kontsevich’s work). In a future post, we’ll illustrate some of G. Forni’s techniques by showing the positivity of the second Lyapunov exponent of the KZ cocycle with respect to Masur-Veech measure . While the fact is certainly a weaker statement than Forni’s theorem , it turns out that it is sufficient to some interesting applications to interval exchange transformations and vertical translation flows. Indeed, using a technical machinery of parameter exclusion strongly based on the fact that , A. Avila and G. Forni were able to show that almost every i.e.t. (not corresponding to “rotations”) and almost every vertical translation flow (on genus translation surfaces) are *weakly mixing*. Here, we say that an i.e.t. corresponds to a rotation if its combinatorial data has the form (mod ). In this case, one can see that the corresponding i.e.t. can be conjugated to a rotation of the circle, and hence it is never weak-mixing. Observe that, in general, weak-mixing property is the “best” dynamical property we can expect: indeed, as it was shown by A. Katok, interval exchange transformations and suspension flows over i.e..t’s with a roof function of bounded variation (e.g., translation flows) are never *mixing*.

In 2007, A. Avila and M. Viana proved the full Kontsevich-Zorich conjecture by studying a discrete-time analog of Kontsevich-Zorich cocycle over the Rauzy-Veech induction. In few words, Avila and Viana showed that the symplectic monoid associated to Rauzy-Veech induction is *pinching* (“it contains matrices with simple spectrum”) and *twisting* (“any subspace can be put into generic position by using some matrix of the monoid”), and they used the pinching and twisting properties to ensure simplicity of Lyapunov spectra. In a certain sense, these conditions (pinching and twisting) are analogues (for deterministic chaotic dynamical systems) of the *strong irreducibility* and *proximality* conditions (sometimes derived from a stronger *Zariski density property*) used by Y. Guivarch and A. Raugi, and I. Goldsheid and G. Margulis to derive simplicity of Lyapunov exponents for random products of matrices.

As the reader can imagine, the Kontsevich-Zorich conjecture has applications to the study of *deviations* of ergodic averages along trajectories of vertical translation flows and interval exchanges transformations. Actually, this was the initial motivation for the introduction of the Kontsevich-Zorich cocycle by A. Zorich and M. Kontsevich.

For the case of vertical translation flows, we begin with a typical vertical translation flow on a translation surface (so that it is uniquely ergodic) and we choose a typical point (so that is defined for every time ), e.g., as in Figure~1 above. For all large enough, let us denote by the homology class obtained by “closing” the piece of (vertical) trajectory with a bounded (usually *small*) segment connecting to . Recall that S. Schwartzman theorem says that

For genus translation surfaces (i.e., flat torii), this is very good and fairly complete result: indeed, it is not hard to see that the *deviation* of from the line spanned by the Schwartzman asymptotic cycle is *bounded*.

For genus translation surfaces, the global scenario gets richer: by doing numerical experiments, what one sees is that the *deviation* of from the line has amplitude with around a certain line. In other words, the deviation of from the Schwartzman asymptotic cycle is not completely random: it occurs along an *isotropic* 2-dimensional plane containing . Again, in genus , this is a “complete” picture in the sense that numerical experiments indicate that the deviation of from is again bounded.

More generally, for arbitrary genus , the numerical experiments indicate that existence of an *asymptotic Lagrangian flag*, i.e., a sequence of isotropic subspaces with and a *deviation spectrum* such that

for every , and

For instance, the reader can see below two figures (extracted from A. Zorich’s survey) showing numerical experiments related to the *deviation phenomenon* or *Zorich phenomenon*discussed above in a genus 3 translation surface. There, we have a slightly different notation for the involved objects: denotes for a convenient choice of , the subspaces correspond to the subspaces , and the numbers correspond to the numbers .

This scenario supported by numerical experimental was made rigorous by A. Zorich using the Kontsevich-Zorich cocycle: more precisely, he proved that the previous statement is true with corresponding to the sum of the Oseledets subspaces associated to the first non-negative exponents of KZ cocycle, and corresponding to the -th Lyapunov exponent of the KZ cocycle with respect to Masur-Veech measure . Of course, to get the *complete* description of the deviation phenomenon (i.e., the fact that , that is, the asymptotic flat is Lagrangian and complete), one needs to know that Kontsevich-Zorich conjecture is true. So, in this sense, A. Zorich’s theorem is a conditional statement depending on Kontsevich-Zorich conjecture.

Closing this subsection, let us mention that a similar scenario of deviations of ergodic averages for i.e.t.’s is true (as proved by A. Zorich in the same 1994 paper), but its precise statement is somewhat technical because we need to talk first about *special Birkhoff sums* (which are Birkhoff sums along trajectories of our initial i.e.t. from a point until its return to special intervals [determined by Rauzy-Veech algorithm]), and then decompose general Birkhoff sums into a sum of relatively few special Birkhoff sums. In particular, we’ll not comment on this here, and we refer the curious reader to A. Zorich‘s original paper and J.-C. Yoccoz survey.

** 1.4. Exponential mixing (and spectral gap of representations) **

Generally speaking, we say that a flow on a space is *mixing* with respect to an invariant probability when the *correlation function* satisfies

for every . Of course, the *mixing* property always implies *ergodicity *of but the converse is not always true (e.g., irrational translation flows on the torus are ergodic but not mixing). However, as we’re going to see in a moment, when the flow is part of a larger action, it is possible to show that ergodicity implies mixing.

More precisely, suppose that we have a action on a space preserving a probability measure , and let be the flow on corresponding to the action of the diagonal subgroup of . In this setting, one has:

Of course, the Teichmüller flow on a connected component of a stratum of the moduli space of Abelian differentials equipped with its natural (Masur-Veech) probability measure is a prototype example of flow verifying the assumptions of the previous proposition.

As we pointed out above, the proof of this result uses knowledge of the representation theory of . In particular, we’ll borrow the facts and even notations used in this post here.

We begin by observing that the action on induces an unitary representation of on . Here, is the Hilbert space of functions of with zero mean. In particular, from the semisimplicity of , we can write as a integral of irreducible unitary representations :

The fact that is -ergodic implies that the action is -ergodic, that is, the trivial representation doesn’t appear in the previous integral decomposition. By Bargmann’s classification, every nontrivial unitary irreducible representation belongs to one of the following three classes (or series): *principal series*, *discrete series* and *complementary series*. See this post for more discussion.

By M. Ratner’s work, we know that, for every and for every with in the *principal or discrete* series,

where is an universal constant. Of course, we’re implicitly using the fact that, by hypothesis, is exactly the action of the diagonal subgroup of on . Also, for every and for every vectors (see this post for more details) with in the *complementary* series, one can find a parameter (related to the eigenvalue of the *Casimir operator*of ) such that

where is a constant depending only on and is the norm of a vector along the direction. In the notation of this post, . Furthermore, can be taken uniform on intervals of the form with .

Putting these informations together (and using the classical fact that vectors are dense), one obtains that as (actually, it goes “exponentially fast” to zero in the sense explained above) for:

- all vectors when belongs to the principal or discrete series;
- a dense subset (e.g., vectors) of vectors when belongs to the complementary series.

Using this (and the integral decomposition ), we conclude that as for a dense subset of vectors . Now, an easy approximation argument shows that as for all . Hence, is -mixing and the proof of Proposition 4 is complete.

Once the Proposition 4 is proved, a natural question concerns the “speed”/“rate” of convergence of to zero (as ). In a certain sense, this question was already answered during the proof of Proposition 4: using Ratner’s results, one can show that converges exponentially fast to zero for all in a dense subet of (e.g., vectors) if and only if the unitary representation has *spectral gap*, i.e., there exists such that, when writing as an integral of unitary irreducible represenations, no in the complementary series has parameter . Actually, it is possible to show that the spectral gap property is equivalent to the nonexistence of *almost invariant vectors*: recall that a representation of a Lie group on a Hilbert space has almost vector when, for all compact subsets and for all , there exists an unit vector such that for all .

In general, it is a hard task to prove the spectral gap property for a given unitary representation. For the case of the unitary representation obtained from the action on a connected component of a stratum of the moduli space of Abelian differentials *equipped with the natural Masur-Veech measure* , A. Avila, S. Gouëzel and J.-C. Yoccoz showed the following theorem:

Theorem 5 (A. Avila, S. Gouëzel, J.-C. Yoccoz)The Teichmüller flow on is exponentially mixing with respect to (in the sense that exponentially as for “sufficiently smooth” ), and the unitary representation has spectral gap.

In the proof of this result, Avila, Gouëzel and Yoccoz proves *firstly* that the Teichmüller geodesic flow (i.e., the action of the diagonal subgroup on the moduli space of Abelian differentials) is *exponentially mixing* with respect to Masur-Veech measure (indeed this is the main result of their paper) and they use a *reverse Ratner estimate* to derive the spectral gap property *from* the exponential mixing (and not the other way around!). Here, the proof of the exponential mixing property with respect to Masur-Veech measure is obtained by delicate (mostly combinatorial) estimates on the so-called *Rauzy-Veech induction*.

More recently, Avila and Gouëzel developed a more geometrical (and less combinatorial) approach to the exponential mixing of *algebraic* -invariant probabilities.

Roughly speaking, an algebraic -invariant measure is a probability measure supported on an *affine suborbifold* of (in the sense that corresponds, in local period coordinates, to affine subspaces in relative homology) such that is absolutely continuous (wrt the Lebesgue measure on the affine subspaces corresponding to in period charts) and its density is locally constant in period coordinates. The class of algebraic -invariant probabilities contains all “known” examples (e.g., Masur-Veech measures and the probabilities supported on the -orbits of Veech surfaces [in particular, square-tiled surfaces]). Actually, an important conjecture in Teichmüller dynamics claims that *all* -invariant probabilities are algebraic. If it is true, this conjecture would provide a non-homogenous counterpart to Ratner’s theorems on unipotent actions in homogenous spaces.

After the celebrated works of K. Calta and C. McMullen, there is a *complete* classification of -invariant measures in genus (i.e., or ). In particular, it follows that such measures are always algebraic (in genus ). Furthermore, it was recently announced by A. Eskin and M. Mirzakhani that the full conjecture is true.

In any case, the result obtained by Avila and Gouëzel is:

Theorem 6 (Avila and Gouëzel)Let be an algebraic -invariant probability, and consider the integral decomposition of the unitary representation into irreducible factors . Then, for any , the representations of the complementary series with parameter appear onlydiscretely(i.e., is finite) and withfinite multiplicity(i.e., for each , is finite). In particular, the Teichmüller geodesic flow is exponentially mixing with respect to .

In a future post, we will highlight some aspects of the proof of this result.

This is all for today’s post! Next time, we’ll discuss a question posed by W. Veech to G. Forni…

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