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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Posted by: matheuscmss | September 3, 2013

Semisimplicity of the Lyapunov spectrum for irreducible cocycles

Alex Eskin and I have just uploaded to ArXiv our paper “Semisimplicity of the Lyapunov spectrum for irreducible cocycles”.

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 {G} be a semisimple Lie group acting on a space {X}. Denote by {\mu} a compactly supported probability measure on {G} and {\nu} a {\mu}-stationary probability on {X}, i.e., {\int_G g_*\nu d\mu(g):=\mu\ast\nu=\nu} (that is, {\nu} is “invariant on {\mu}-average” under push-forwards by elements of {G}).

A linear cocycle is {A: G\times X\rightarrow SL(L)} where {L} is a real finite-dimensional vector space. For our purposes, we will assume that the matrices {A(g,x)} are bounded for any {g} in the support of {\mu}.

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

Let {\Omega=G^{\mathbb{N}}} and denote by {T:\Omega\times X\rightarrow\Omega\times X} be the natural shift map {T(u,x)=(\sigma(u),u_1(x))} where {u=(u_1, u_2,\dots)\in\Omega}. Observe that, by {\mu}-stationarity of {\nu}, the product probability measure {\beta\times\nu} is {T}-invariant where {\beta=\mu^{\mathbb{N}}}. For the sake of simplicity, we will assume that the {\mu}-stationary measure {\nu} is ergodic in the sense that {\beta\times\nu} is {T}-ergodic.

In this language, the orbits of the map {F_A:\Omega\times X\times SL(L)\rightarrow \Omega\times X\times SL(L)} given by

\displaystyle F_A(u,x,z)=(T(u,x), A(u_1,x)z)

are modeling random products of the matrices {A(g,x)} along random walks in {G}. Indeed, this is clearly seen through the formula:

\displaystyle F^n_A(u,x,Id)=(T^n(u,x), A(u_n,u_{n-1}\dots u_1(x))\dots A(u_1,x))=:(T^n(u,x), A^n(u,x))

In this context, Oseledets theorem applied to {F_A} ensures the existence of a collection of numbers {\lambda_1>\dots>\lambda_k} with multiplicities {m_1,\dots, m_k} called Lyapunov exponents and, at {\beta\times\nu}-almost every point {(u,x)\in\Omega\times X}, a Lyapunov flag

\displaystyle \{0\}=V_{k+1}\subset V_k(u,x)\subset\dots\subset V_1(u,x)=L

such that {V_i(u,x)} is a subspace of dimension {m_i+\dots+m_k} and

\displaystyle \lim\limits_{n\rightarrow\infty}\frac{1}{n}\log\|A^n(u,x)\vec{p}\|=\lambda_i

whenever {\vec{p}\in V_i(u,x)\setminus V_{i+1}(u,x)}.

The collection of Lyapunov exponents and Lyapunov flags of {A(.,.)} (or, more precisely, {F_A}) is called the Lyapunov spectrum of {A(.,.)}.

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 {SL(L)} or in the symplectic group {Sp(L)}) ensuring the simplicity of the Lyapunov spectrum (that is, the multiplicities {m_i} are all equal to {1}). More recently, Avila-Viana gave an alternative set of geometrical conditions (“pinching” and “twisting”) ensuring simplicity for cocycles {A} taking values in both {SL(L)} and {Sp(L)}, 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 {O(p,q)}, {U_{\mathbb{C}}(p,q)} and {U_{\mathbb{H}}(p,q)} (resp.) of real, complex and quaternionic matrices (resp.) preserving an indefinite form of signature {(p,q)} on {L}.

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 {SL(2,\mathbb{R})}-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 {F_A} (or, equivalently, replacing the cocycle {A(.,)} by {C(g(x)) A(g,x) C(x)^{-1}} for some adequate measurable map {C:X\rightarrow SL(L)}), the cocycle {A} is block-conformal, i.e.,

\displaystyle A(g,x)=\left(\begin{array}{ccc}c_1 & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & c_m\end{array}\right)

where {c_l}‘s are conformal (that is, {c_l} belongs to an orthogonal group after multiplication by an adequate constant). Equivalently, {A} has semisimple Lyapunov spectrum if we can write the quotients {V_i(u,x)/V_{i+1}(u,x)} between consecutive subspaces of the Lyapunov flag as

\displaystyle V_{i}(u,x)/V_{i+1}(u,x) = \bigoplus\limits_{j=1}^{n_i} E_{ij}(u,x) \ \ \ \ \ (1)

where each {E_{ij}(u,x)} admits a non-degenerate quadratic form {\langle.,.\rangle_{ij,u,x}} such that for all {\vec{p}, \vec{q}\in E_{ij}(u,x)} and for all {n\in\mathbb{N}} one has

\displaystyle \langle A^n(u,x)\vec{p}, A^n(u,x)\vec{q} \rangle_{ij,T^n(u,x)} = e^{\lambda_{ij}(u,x,n)}\langle\vec{p},\vec{q}\rangle_{ij,u,x}

with {\lambda_{ij}:\Omega\times X\times\mathbb{N}\rightarrow \mathbb{R}} satisfying the cocycle relation

\displaystyle \lambda_{ij}(u,x,m+1)=\lambda_{ij}(T^{m-1}(u,x),1)+\lambda_{ij}(u,x).

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

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 {A(.,.)} whose algebraic hull is {SL(L)} is explained in Appendix C of Eskin-Mirzakhani’s paper. Here, by algebraic hull we mean the smallest {\mathbb{R}}-algebraic subgroup {\textbf{H}} such that {C(g(x)) A(g,x) C(x)^{-1}\in\textbf{H}} for some measurable (conjugation) map {C: X\rightarrow SL(L)} (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 {\textbf{H}} (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 {\lambda_1} is associated to a single conformal block in the sense that the decomposition in (1) is trivial (i.e., {n_1=1}) for {V_1(u,x)/V_2(u,x)}.

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 {\textbf{H}} by {SL(L)} (or {Sp(L)}, {U_{\mathbb{C}}(p,q)}) 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 {\textbf{H}} on stationary measures supported on a flag variety {\textbf{H}/P_I} (where {P_I} is a certain parabolic subgroup corresponding to some subset {I} 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 {A(.,.)}. Firstly, by analyzing (in Section 2 of our preprint) the action of elements of the algebraic hull {\textbf{H}} on adequate flag varieties {\textbf{H}/P_{I}} (for some choices {I} of subsets of simple roots), we show that {A(.,.)} can be conjugated to take its values in a certain parabolic subgroup {P_{I'}} (cf. Proposition 3.2 of our preprint). In a certain sense, this information seems of little value because, very roughly speaking, the fact that {A(.,.)} takes values in a parabolic subgroup essentially amounts to say that we can put {A(.,.)} in the form

\displaystyle A(g,x)=\left(\begin{array}{ccc}c_1 & \ast & \ast \\ 0 & \ddots & \ast \\ 0 & 0 & c_m\end{array}\right)

and this is certainly not the desired block conformality property (as this last property means that all {\ast} entries above are all zero). Nonetheless, we combine the information {A(.,.)} 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 {G}) to deduce that {A(.,.)} 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 {A(.,.)} is block-conformal (i.e., all {\ast}‘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 {\textbf{H}} of the strongly irreducible cocycle {A(.,.)} on the real finite-dimensional vector space {L} has multiplicity {1}.

Posted by: matheuscmss | August 20, 2013

Workshop on Combinatorics, Number Theory and Dynamical Systems

I’m extremely happy to announce that the Workshop on Combinatorics, Number Theory and Dynamical Systems started today at IMPA, Rio de Janeiro, Brazil (as part of the activities of a thematic semester from August to on Dynamical Systems mainly organized by Carlos (Gugu) Moreira, Enrique Pujals and Marcelo Viana).

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”).

 

The recent breakthrough article of A. Eskin and M. Mirzakhani sheds some light about the geometric structure of {SL(2,\mathbb{R})}-invariant probability measures on moduli spaces of Abelian differentials. In a nutshell, they showed the following analog of the celebrated Ratner’s measure classification theorem in the non-homogenous setting of moduli spaces of Abelian differentials: any ergodic {SL(2,\mathbb{R})}-invariant probability measure on these moduli spaces is an affine measure fully supported on some affine suborbifold.

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 {SL(2,\mathbb{R})}-invariant subbundles of the Hodge bundle.

It is interesting to point out that, while symplectic {SL(2,\mathbb{R})}-invariant subbundles of the Hodge bundle occur in several known examples (see, e.g., these articles here), the existence of some example of isotropic {SL(2,\mathbb{R})}-invariant subbundle is not so clear.

Indeed, the question of the existence of non-trivial isotropic {SL(2,\mathbb{R})}-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 {SL(2,\mathbb{R})}-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 {(X,\omega)} of genus {15} with {512} squares such that the Hodge bundle over the {SL(2,\mathbb{R})}-orbit of {(X,\omega)} has non-trivial isotropic {SL(2,\mathbb{R})}-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 {(X,\omega)} (leaving a few details to the our note with Gabi).

Read More…

Posted by: matheuscmss | August 9, 2013

Computation of the Poisson boundary of a lattice of SL(2,R)

In this previous post, we discussed some results of Furstenberg on the Poisson boundaries of lattices of {SL(n,\mathbb{R})} (mostly in the typical low-dimensional cases {n=2} and/or {n=3}). In particular, we saw that it is important to know the Poisson boundary of such lattices in order to be able to distinguish between them.

More precisely, using the notations of this post (as well as of its companion), we mentioned that a lattice {\Gamma} of {SL(n,\mathbb{R})} can be equipped with a probability measure {\mu} such that the Poisson boundary of {(\Gamma,\mu)} coincides with the Poisson boundary {(B_n, m_{B_n})} of {SL(n,\mathbb{R})} equipped with any spherical measure (cf. Theorem 13 of this post). Then, we sketched the construction of the probability measure {\mu} in the case of a cocompact lattice {\Gamma} of {SL(2,\mathbb{R})}, and, after that, we outlined the proof that {(B_n, m_{B_n})} is a boundary of {(\Gamma,\mu)} in the cases {n=2} and {3}.

However, we skipped a proof of the fact that {(B_n, m_{B_n})} is the Poisson boundary of {(\Gamma, \mu)} by postponing it possibly to another post. Today our plan is to come back to this point by showing that {(B_2, m_{B_2})} is the Poisson boundary of {(\Gamma,\mu)}.

More concretely, we will show the following statement due to Furstenberg. Let {\Gamma} be a cocompact lattice of {SL(2,\mathbb{R})}. As we saw in this previous post (cf. Proposition 14), one can construct a probability measure {\mu} on {\Gamma} such that

  • (a) {\mu} has full support: {\mu(\{\gamma\})>0} for all {\gamma\in\Gamma},
  • (b) {m_{B_2}} is {\mu}-stationary: {\mu\ast m_{B_2}=m_{B_2}},
  • (c) the {\log}-norm function is {\mu}-integrable: {\sum\limits_{\gamma\in\Gamma} \mu(\gamma)\Lambda(\gamma)<\infty}.

Here, we recall (for the sake of convenience of the reader) that: {B_2\simeq\mathbb{P}^1} is the “complete flag variety” of {\mathbb{R}^2} or, equivalently, {B_2=SL(2,\mathbb{R})/P_2} where {P_2} is the subgroup of upper-triangular matrices, {m_{B_2}} is the Lebesgue (probability) measure and

\displaystyle \Lambda(g):=d(g(0),0)=\log\frac{1+|g(0)|}{1-|g(0)|}

where {g\in SL(2,\mathbb{R})} acts on Poincaré’s disk {\mathbb{D}} via Möebius transformations (as usual) and {d} denotes the hyperbolic distance on Poincaré’s disk {\mathbb{D}}.

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

Theorem 1 Let {\Gamma} be a cocompact lattice of {SL(2,\mathbb{R})} and denote by {\mu} any probability measure on {\Gamma} satisfying the conditions in items (a), (b) and (c) above. Then, the Poisson boundary of {(\Gamma,\mu)} is {(B_2, m_{B_2})}.

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

Read More…

Alex Wright and I have just upload to ArXiv our paper Hodge-Teichmüller planes and finiteness results for Teichmüller curves.

As the title of the paper indicates, this article concerns finiteness results for certain classes of Teichmüller curves (i.e., closed {SL(2,\mathbb{R})}-orbits in the moduli spaces of Abelian differentials). For example, one of the results in this paper is the finiteness of algebraically primitive Teichmüller curves in minimal strata of prime genus \geq 3.

In fact, some parts of this paper were previously discussed in this blog here, and, for this reason, we will not make further comments on the contents of this article today.

Instead, let us take the opportunity to briefly discuss a “deleted scene” of this paper.

More concretely, at some stage of the paper, Alex and I need to know that there are Abelian differentials/translation surfaces with “rich monodromy” (whatever this means) in each connected component of every stratum of genus {g\geq 3}.

In this direction, we ensure the existence of such translation surfaces by an inductive argument going as follows.

Starting with an hypothetical connected component {\mathcal{H}} of genus {g\geq 3} containing only translation surfaces with “poor monodromy” ({g-1} orthogonal Hodge-Teichmüller planes in the notation of the paper), we follow and adpat some arguments of Kontsevich-Zorich paper on the classification of connected components of strata (cf. Section 5 of our paper) and we work a little bit with the Deligne-Mumford compactification of moduli spaces (cf. Section 6 of our paper) to show that the “poorness of monodromy” property passes down to all translations surfaces in both connected components {\mathcal{H}(4)^{hyp}} and {\mathcal{H}(4)^{odd}}. Here, very roughly speaking, the basic idea is that, if we “degenerate” (e.g., pinch off a curve) in a careful way the translation surfaces in {\mathcal{H}}, we can decrease the genus without loosing the “poorness of monodromy” property, so that, if we keep degenerating, then we will find ourselves with plenty of genus {3} translation surfaces with poor monodromy.

However, it is easy to see that the property that all translation surfaces in {\mathcal{H}(4)^{hyp}} and {\mathcal{H}(4)^{odd}} have “poor monodromy” is false: indeed, in Section 4 of our paper, we exhibit explicitely two translation surfaces {M_{\ast}\in \mathcal{H}(4)^{odd}} and {M_{\ast\ast}\in\mathcal{H}(4)^{hyp}} with “rich monodromy”. Thus, the hypothetical connected component {\mathcal{H}} can not exist.

As it turns out, before thinking about this argument based on Kontsevich-Zorich classification of connected components and the features of Deligne-Mumford compactification, Alex and I thought of using the following simple-minded argument. One can check that the “richness of monodromy” property passes through finite branched coverings. Hence, it suffices to produce for each connected component {\mathcal{H}} an explicit finite branched cover of one of the translation surfaces {M_{\ast}} or {M_{\ast\ast}} lying in {\mathcal{H}} to deduce that all connected components of all strata possess translation surfaces with “rich monodromy”.

Remark 1 The strategy in the previous paragraph is very natural: for instance, after reading a preliminary version of our paper, Giovanni Forni asked us if we could not make the arguments in Sections 5 and 6 of our paper (related to Kontsevich-Zorich classification of connected components of strata and Deligne-Mumford compactification) more elementary by taking finite branched covers to produce explicit translation surfaces with rich monodromy on each connected component of every stratum. As we will see below, this does not quite work to fully recover the statements in Section 5 of our paper, but it allows to obtain at least part of our statements.

In particular, around November/December 2012, Alex and I started looking at finite branched covers of {M_{\ast}} and {M_{\ast\ast}} “hitting” all connected components of all strata. Unfortunately, this strategy does not work as well as one could imagine: firstly, there are restrictions on the genera of strata we can reach using finite branched covers (thanks to Riemann-Hurwitz formula), and, secondly, it is not so easy to figure out in what connected component our finite branched cover lives.

Nevertheless, this elementary strategy permits to deduce the following proposition (allowing to deduce partial versions of the statements in Section 5 of our paper). Let {M_{\ast}\in \mathcal{H}(4)^{odd}} and {M_{\ast\ast}\in\mathcal{H}(4)^{hyp}} be the square-tiled surfaces with “rich monodromy” constructed in our paper (see also this post), i.e., the square-tiled surfaces below

MW-H4odd

M_{\ast}\in\mathcal{H}(4)^{odd}

MW-H4hyp

M_{\ast\ast}\in\mathcal{H}(4)^{hyp}

associated to the pairs of permutations {h_{\ast}=(1)(2,3)(4,5,6)} and {v_{\ast}=(1,4,2)(3,5)(6)}, and {h_{\ast\ast}=(1)(2,3)(4,5,6)} and {v_{\ast\ast}=(1,2)(3,4)(5)}.

Remark 2 Here, as it is usual in this theory, we are constructing square-tiled surfaces from a pair of permutations {h, v\in S_N} on {N} elements by taking {N} unit squares and gluing them (by translations) so that {h(i)} is the square to the right of the square {i} and {v(i)} is the square on the top of the square {i}.

Proposition 1 For each {d\geq 3} odd, there exists {M_{\ast}(d)} a finite branched cover of {M_{\ast}} of degree {d} in the odd connected component {\mathcal{H}(5d-1)^{odd}} of the minimal stratum {\mathcal{H}(5d-1)} (of Abelian differentials with a single zero of order {5d-1}).

Also, there exists {M_{\ast\ast}(3)} a finite branched cover of {M_{\ast\ast}} of degree {3} in the hyperelliptic connected component {\mathcal{H}(14)^{hyp}} of the minimal stratum {\mathcal{H}(14)} (of Abelian differentials with a single zero of order {14}).

Finally, for each {d\geq 5}, there exists {M_{\ast\ast}(d)} a finite branched cover of {M_{\ast\ast}} of degree {d} in the even connected component {\mathcal{H}(5d-1)^{even}} of the minimal stratum {\mathcal{H}(5d-1)} (of Abelian differentials with a single zero of order {5d-1}).

Remark 3 In this statement, the nomenclature “hyperelliptic”, “even” and “odd” refers to the invariants introduced by Kontsevich and Zorich to distinguish between the connected components of strata. We will briefly review these notions below (as we will need them to prove Proposition 1.

We will dedicate the remainder of this post to outline the construction of the translation surfaces {M_{\ast}(d)} and {M_{\ast\ast}(d)} satisfying the conclusions of Proposition 1. In particular, we will divide the discussion into three sections: in the next one, we will quickly review the invariants introduced by Kontsevich-Zorich to classify connected components of strata, and in the last two sections we will construct {M_{\ast}(d)} and {M_{\ast\ast}(d)} respectively.

Read More…

Last time we introduced Poisson boundaries hoping to use them to distinguish between lattices of {SL(2,\mathbb{R})} and {SL(n,\mathbb{R})}, {n\geq 3}. More precisely, we followed Furstenberg to define and construct Poisson boundaries as a tool that would allow to prove the following statement:

Theorem 1 (Furstenberg (1967)) A lattice of {SL(2,\mathbb{R})} can’t be realized as a lattice in {SL(n,\mathbb{R})} for {n\geq 3} (or, in the language introduced in the previous post, {SL(n,\mathbb{R})}, {n\geq 3}, can’t envelope a discrete group enveloped by {SL(2,\mathbb{R})}).

Here, we recall that, very roughly speaking, these Poisson boundaries {(B,\nu)} were certain “maximal” topological objects attached to locally compact groups with probability measures {(G,\mu)} in such a way that the points in the boundary {B} were almost sure limits of {\mu}-random walks on {G} and the probability measure {\nu} was a {\mu}-stationary measure giving the distribution at which {\mu}-random walks hit the boundary.

For this second (final) post, we will discuss (below the fold) some examples of Poisson boundaries and, after that, we will sketch the proof of Theorem 1.

Remark 1 The basic references for this post are the same ones from the previous post, namely, Furstenberg’s survey, his original articles and A. Furman’s survey.

Read More…

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