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

 

Theorem 1 (Burns-Masur-Wilkinson) Let {N} be the quotient {N=M/\Gamma} of a contractible, negatively curved, possibly incomplete, Riemannian manifold {M} by a subgroup {\Gamma} of isometries of {M} acting freely and properly discontinuously. Denote by {\overline{N}} the metric completion of {N} and {\partial N:=\overline{N}-N} the boundary of {N}.Suppose that:

  • (I) the universal cover {M} of {N} is geodesically convex, i.e., for every {p,q\in M}, there exists an unique geodesic segment in {M} connecting {p} and {q}.
  • (II) the metric completion {\overline{N}} of {(N,d)} is compact.
  • (III) the boundary {\partial N} is volumetrically cusplike, i.e., for some constants {C>1} and {\nu>0}, the volume of a {\rho}-neighborhood of the boundary satisfies

    \displaystyle \textrm{Vol}(\{x\in N: d(x,\partial N)<\rho\})\leq C \rho^{2+\nu}

    for every {\rho>0}.

  • (IV) {N} has polynomially controlled curvature, i.e., there are constants {C>1} and {\beta>0} such that the curvature tensor {R} of {N} and its first two derivatives satisfy the following polynomial bound

    \displaystyle \max\{\|R(x)\|,\|\nabla R(x)\|,\|\nabla^2 R(x)\|\}\leq C d(x,\partial N)^{-\beta}

    for every {x\in N}.

  • (V) {N} has polynomially controlled injectivity radius, i.e., there are constants {C>1} and {\beta>0} such that

    \displaystyle \textrm{inj}(x)\geq (1/C) d(x,\partial N)^{\beta}

    for every {x\in N} (where {inj(x)} denotes the injectivity radius at {x}).

  • (VI) The first derivative of the geodesic flow {\varphi_t} is polynomially controlled, i.e., there are constants {C>1} and {\beta>0} such that, for every infinite geodesic {\gamma} on {N} and every {t\in [0,1]}:

    \displaystyle \|D_{\stackrel{.}{\gamma}(0)}\varphi_t\|\leq C d(\gamma([-t,t]),\partial N)^{\beta}

Then, the Liouville (volume) measure {m} of {N} is finite, the geodesic flow {\varphi_t} on the unit cotangent bundle {T^1N} of {N} is defined at {m}-almost every point for all time {t}, and the geodesic flow {\varphi_t} is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.

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

 

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

 

Theorem 2 Under the assumptions (II), (III) and (VI) in Theorem 1 above, the geodesic flow {\varphi_t} is non-uniformly hyperbolic: more concretely, there exists a subset {\Lambda_0\subset T^1N} of full {m}-measure such that the {D\varphi_t}-invariant splitting

\displaystyle T_vT^1N=E^s(v)\oplus E^0(v)\oplus E^u(v)

into the flow direction {E^0(v)=\mathbb{R}\dot{\varphi}(v)} and the spaces {E^s(v)} and {E^u(v)} of stable and unstable Jacobi fields along {\gamma(t)=\varphi_t(v)} have the property that

\displaystyle 0<\lim\limits_{t\rightarrow\infty}\frac{1}{t}\log\|D_v\varphi_t(\xi^u)\|<\infty \quad \textrm{and} \quad -\infty<\lim\limits_{t\rightarrow\infty}\frac{1}{t}\log\|D_v\varphi_t(\xi^s)\|<0

for all {\xi^u\in E^u(v)-\{0\}} and {\xi^s\in E^s(v)-\{0\}}.

 

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

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

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Posted by: matheuscmss | May 23, 2014

What is … the Kontsevich-Zorich cocycle?

In this post (with title inspired by the “What is …” column in Notices of the AMS), I would like to record some conversations I had with Jean-Christophe Yoccoz (mostly by the time we wrote our joint paper with David Zmiaikou) about a little technical issue arising when one tries to see the so-called Kontsevich-Zorich cocycle as a linear cocycle (in the usual sense of Dynamical Systems) over the Teichmüller flow (and/or {SL(2,\mathbb{R})}-action) on moduli spaces of translation surfaces.

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

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

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

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Last time, we reduced the proof of Burns-Masur-Wilkinson theorem on the ergodicity (and mixing) of the Weil-Petersson geodesic flow to a certain estimate for the first derivative of a geodesic flow on negatively curved manifolds (cf. Theorem 11 in this post) and Burns-Masur-Wilkinson ergodicity criterion for geodesic flows on some negatively curved manifolds (cf. Theorem 1 in this post).

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

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It is a pleasure to announce in this short post the following two interesting forthcoming events in Dynamical Systems:

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

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

 

Last Wednesday, Jean-Christophe Yoccoz gave a talk (in French) entitled “Regularité holdérienne des solutions de l’équation cohomologique pour les échanges d’intervalle”.

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.

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Posted by: matheuscmss | February 5, 2014

On the continuity of Lyapunov spectrum for random products

Last week, the conference “Random walks on groups” took place at IHP as part of the activities of a trimester on random walks and asymptotic geometry of groups (organized by Indira Chatterji, Anna Erschler, Vadim Kaimanovich, and Laurent Saloff-Coste) from January to March 2014.

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 1 A 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 explanations concerning “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. 

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In the first post of this series, we planned to discuss in the third and fourth posts the proof of the following ergodicity criterion for geodesic flows in incomplete negatively curved manifolds of Burns-Masur-Wilkinson:

Theorem 1 (Burns-Masur-Wilkinson) Let {N} be the quotient {N=M/\Gamma} of a contractible, negatively curved, possibly incomplete, Riemannian manifold {M} by a subgroup {\Gamma} of isometries of {M} acting freely and properly discontinuously. Denote by {\overline{N}} the metric completion of {N} and {\partial N:=\overline{N}-N} the boundary of {N}.Suppose that:

  • (I) the universal cover {M} of {N} is geodesically convex, i.e., for every {p,q\in M}, there exists an unique geodesic segment in {M} connecting {p} and {q}.
  • (II) the metric completion {\overline{N}} of {(N,d)} is compact.
  • (III) the boundary {\partial N} is volumetrically cusplike, i.e., for some constants {C>1} and {\nu>0}, the volume of a {\rho}-neighborhood of the boundary satisfies

    \displaystyle \textrm{Vol}(\{x\in N: d(x,\partial N)<\rho\})\leq C \rho^{2+\nu}

    for every {\rho>0}.

  • (IV) {N} has polynomially controlled curvature, i.e., there are constants {C>1} and {\beta>0} such that the curvature tensor {R} of {N} and its first two derivatives satisfy the following polynomial bound

    \displaystyle \max\{\|R(x)\|,\|\nabla R(x)\|,\|\nabla^2 R(x)\|\}\leq C d(x,\partial N)^{-\beta}

    for every {x\in N}.

  • (V) {N} has polynomially controlled injectivity radius, i.e., there are constants {C>1} and {\beta>0} such that

    \displaystyle \textrm{inj}(x)\geq (1/C) d(x,\partial N)^{\beta}

    for every {x\in N} (where {inj(x)} denotes the injectivity radius at {x}).

  • (VI) The first derivative of the geodesic flow {\varphi_t} is polynomially controlled, i.e., there are constants {C>1} and {\beta>0} such that, for every infinite geodesic {\gamma} on {N} and every {t\in [0,1]}:

    \displaystyle \|D_{\stackrel{.}{\gamma}(0)}\varphi_t\|\leq C d(\gamma([-t,t]),\partial N)^{\beta}

Then, the Liouville (volume) measure {m} of {N} is finite, the geodesic flow {\varphi_t} on the unit cotangent bundle {T^1N} of {N} is defined at {m}-almost every point for all time {t}, and the geodesic flow {\varphi_t} is non-uniformly hyperbolic (in the sense of Pesin’s theory) and ergodic.

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

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Posted by: matheuscmss | December 23, 2013

Rates of mixing of Weil-Petersson geodesic flows

Keith Burns, Howard Masur, Amie Wilkinson and I have just upload to ArXiv our research announcement note “Rates of mixing of Weil-Petersson geodesic flows”.

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 \mathcal{M}_{g,n} of surfaces of genus g\geq 0 and n\geq 0 punctures with 3g-3+n>1, 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 \mathcal{M}_{0,4} and once-punctured torii  \mathcal{M}_{1,1} (that is, in the cases when 3g-3+n=1).

As we explain in our note, the speed of mixing of the WP flow on the unit cotangent bundle of \mathcal{M}_{g,n} is polynomial at most when 3g-3+n>1 because the (strata of the) boundary of the moduli spaces \mathcal{M}_{g,n} (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 3g-3+n>1.

More concretely, using this geometrical information on the WP metric near the boundary of \mathcal{M}_{g,n}, 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 \varepsilon>0, one can produce a subset of volume \approx \varepsilon^8 of vectors leading to WP geodesics traveling in the \varepsilon-thin part of \mathcal{M}_{g,n}(= \varepsilon-neighborhood of the boundary of \mathcal{M}_{g,n}) for a time \geq 1/\varepsilon. In other words, there is a subset of volume \approx \varepsilon^8 of geodesics taking time 1/\varepsilon at least to visit the \varepsilon-thick part of \mathcal{M}_{g,n} (= complement of the \varepsilon-thin part of \mathcal{M}_{g,n}). Hence, there are certain non-negligible (volume \approx\varepsilon^8) subsets of the \varepsilon-thin part of \mathcal{M}_{g,n} taking longer and longer (time 1/\varepsilon at least) to mix with the \varepsilon-thick part of \mathcal{M}_{g,n} as \varepsilon\to 0, 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 \mathcal{M}_{g,n} (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 \mathcal{M}_{0,4} of four-times punctured spheres and \mathcal{M}_{1,1} 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 \mathcal{M}_{0,4} and \mathcal{M}_{1,1} 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 \mathcal{M}_{0,4} and \mathcal{M}_{1,1} looks like the metric of the surface of revolution of the profile \{y=x^3\}. 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 \varepsilon-thick part in time \leq 1. Since the WP metric has uniformly bounded negative curvature in the \varepsilon-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 \mathcal{M}_{0,4} and \mathcal{M}_{1,1} converges to -\infty 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).

Today we will define the Weil-Petersson (WP) metric on the cotangent bundle of the moduli spaces of curves and, after that, we will see that the WP metric satisfies the first three items of the ergodicity criterion of Burns-Masur-Wilkinson (stated as Theorem 3 in the previous post).

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.

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Posted by: matheuscmss | November 6, 2013

Dynamics of the Weil-Petersson flow: Introduction

Boris Hasselblatt and Françoise Dal’bo are organizing the event “Young mathematicians in Dynamical Systems” at CIRM (Luminy/Marseille, France) from November 25 to 29, 2013.

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 {\mathcal{M}_{g,n}} of curves (Riemann surfaces) of genus {g\geq 1} with {n\geq 0} 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 {SL(2,\mathbb{R})} action is not part of an {SL(2,\mathbb{R})} 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 {\mu_{T}} ergodic for the Liouville measure {\mu_{WP}}
(j) metric entropy {0<h(\mu_T)<\infty} metric entropy {0<h(\mu_{WP})<\infty}
(k) exponential rate of mixing mixing at most polynomial (in genus {g\geq2})

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 {C^1} but not {C^2} 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 {t\in\mathbb{R}} 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 {t\in\mathbb{R}} 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 {SL(2,\mathbb{R})}-action while the latter corresponds to the action of the diagonal subgroup {g_t=\textrm{diag}(e^t,e^{-t})} of a natural {SL(2,\mathbb{R})}-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 {SL(2,\mathbb{R})}-action makes its dynamics very rich: for instance, once one shows that the Teichmüller flow is ergodic (with respect to some {SL(2,\mathbb{R})}-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 {-\infty} or {0} 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 {\mathcal{M}_{1,1}} (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 {\mathcal{M}_{g,n}} (for any {g\geq 1}, {n\geq 1}) 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 {T^1\mathcal{M}_{1,1}} of the moduli space {\mathcal{M}_{1,1}} of once-punctured tori: they showed that if the first two derivatives of the WP flow on {T^1\mathcal{M}_{1,1}} 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 {T^1\mathcal{M}_{g,n}}of {\mathcal{M}_{g,n}} is ergodic (for any {g\geq 1}, {n\geq 1}) with respect to the Liouville measure {\mu_{WP}} 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 {h(\mu_{WP})} 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 {\psi^t} is the speed of convergence of the correlations functions {C_t(f,g):=\int f\cdot g\circ\psi^t - \left(\int f\right)\left(\int g\right)} to {0} as {t\rightarrow\infty} (for “reasonably smooth” observables {f} and {g}), that is, the speed of {\psi^t} to mix distinct regions of the phase space (such as the supports of the observables {f} and {g}).

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 {T^1\mathcal{M}_{g,n}} is

  • at most polynomial for {g\geq 2} and
  • rapid (super-polynomial) for {g=1}, {n=1}.

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.

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