Posted by: matheuscmss | November 15, 2014

On the ergodicity of billiards in non-rational polygons

A couple of days ago (on November 12th, 2014 to be more precise), Giovanni Forni gave a talk at the “flat seminar / séminaire plat” on the ergodicity of billiards on non-rational polygons, and, by following the suggestion of two friends, I will transcript in this post my notes from Giovanni’s talk.

[Update (November 20, 2014): Some phrases near the statement of Theorem 3 below were edited to correct an inaccuracy pointed out to me by Giovanni.]

Let {\mathcal{P}\subset \mathbb{R}^2} be a polygon with {d+1} sides and denote by {\theta_1, \dots, \theta_{d+1}} its interior angles.

The billiard flow associated to {\mathcal{P}} is the following dynamical system. A point-particle in {\mathcal{P}} follows a linear trajectory with unit speed until it hits the boundary of {\mathcal{P}}. At such an instant, the point-particle is reflected by the boundary of {\mathcal{P}} (according to the usual laws of a specular reflection) and then it follows a new linear trajectory with unit speed. (Of course, this definition makes no sense at the corners of {\mathcal{P}}, and, for this reason, we leave the billiard flow undefined at any orbit going straight into a corner)

The phase space of the billiard flow is naturally identified with the three-dimensional manifold {\mathcal{P}\times S^1}: indeed, we need an element of {\mathcal{P}} to describe the position of the particle and an element of the unit circle {S^1\subset \mathbb{R}^2} to describe the velocity vector of the particle.

Alternatively, the billiard flow associated to {\mathcal{P}} can be interpreted as the geodesic flow on a sphere {S^2} with a flat metric and {(d+1)} conical singularities (whose cone angles are {2\theta_1, \dots, 2\theta_{d+1}}) with non-trivial holonomy (see Section 2 of Zorich’s survey): roughly speaking, one obtains this flat sphere with conical singularities by taking two copies of {\mathcal{P}} (one on the top of the other), gluing them along the boundaries, and by thinking of a billiard flow trajectory on {\mathcal{P}} as a straight line path going from one copy of {\mathcal{P}} to the other at each reflection.

This interpretation shows us that billiard flows on polygons are a particular case of geodesic flows {\{G_t\}} on the unit tangent bundle {S(M-\Sigma)} of compact flat surfaces {M} whose subsets {\Sigma} of conical singularities were removed.

Remark 1 In the case of a rational polygon {\mathcal{P}} (i.e., {\theta_1, \dots, \theta_{d+1}} are rational multiples of {\pi}), it is often a better idea (see this survey of Masur and Tabachnikov) to take several copies of {\mathcal{P}} obtained by applying the finite group generated by the reflections through the sides of {\mathcal{P}} and then glue by translation the pairs of parallel sides of the resulting figure. In this way, one obtains that the billiard flow associated to {\mathcal{P}} is equivalent to translation (straightline) flow on a translation surface (an object that has trivial holonomy and, hence, is more well-behaved that a flat metric on {S^2} with conical singularities) and this partly explains why the Ergodic Theory of billiards on rational polygons is well-developed. However, let us not insist on this point here because in what follows we will be mostly interested in billiard flows on irrational polygons.

A basic problem concerning the dynamics of billiards flows on polygons, or, more generally, geodesic flows on flat surfaces with conical singularities is to determine whether such a dynamical system is ergodic.

In view of Remark 1, we can safely skip the case of rational polygons: indeed, this setting one can use the relationship to translation surfaces to give a satisfactory answer to this problem (see the survey of Masur and Tabachnikov for more explanations). So, from now on, we will focus on billiard flows associated to non-rational polygons.

Kerckhoff, Masur and Smillie proved in 1986 that the billiard flow is ergodic for a {G_{\delta}}-dense subset of polygons. Their idea is to consider the {G_{\delta}}-dense subset of “Liouville polygons” admitting fast approximations by rational polygons (i.e., the subset of polygons whose interior angles admit fast approximations by rational multiples of {\pi}). Because the ergodicity of the billiard flow on rational polygons is well-understood, one can hope to “transfer” this information from rational polygons to any “Liouville polygon”.

Remark 2 The {G_{\delta}}-dense subset of polygons constructed by Kerckhoff, Masur and Smillie has zero measure: indeed, this happens because they require the angles {\theta_1,\dots, \theta_{d+1}} to be “Liouville” (i.e., admit fast approximations by rational multiples of {\pi}), and, as it is well-known, the subset of Liouville numbers has zero Lebesgue measure.

A curious feature of the argument of Kerckhoff, Masur and Smillie is that it is hard to extract any sort of quantitative criterion. More precisely, it is difficult to quantify how fast the quantities {\theta_1/\pi, \dots, \theta_{d+1}/\pi} must be approximated by rationals in order to ensure that the ergodicity of the billiard flow on the corresponding polygon. This happens because the genera of translation surfaces associated to the rational polygons approximating {\theta_1,\dots,\theta_{d+1}} usually tend to infinity and it is a non-trivial problem to control the ergodic properties of translation flows on families of translation surfaces whose genera tend to infinity.

Nevertheless, Vorobets obtained in 1997 (by other methods) a quantitative version of Kerckhoff, Masur and Smillie by showing the ergodicity of the billiard flow on a polygon {\mathcal{P}} whose interior angles {\theta_1,\dots,\theta_{d+1}} verify the following fast approximation property: there exist arbitrarily large natural numbers {N\in \mathbb{N}} such that

\displaystyle \left|\theta_{1}-\pi \frac{p_1}{q_1}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}, \dots, \left|\theta_{d+1}-\pi \frac{p_{d+1}}{q_{d+1}}\right|<\left(2^{2^{2^{2^N}}}\right)^{-1}

for some rational numbers {p_i/q_i\in\mathbb{Q}}, {i=1,\dots, d+1}, with denominators {q_i\leq N}, {i=1,\dots, N}.

In summary, the works of Kerckhoff-Masur-Smillie and Vorobets allows to solve the problem of ergodicity of the billiard flow on Liouville polygons.

Of course, this scenario motivates the question of ergodicity of billiard flows on Diophantine polygons (i.e., the “complement” of Liouville polygons consisting of those {\mathcal{P}} which are badly approximated by rational polygons).

In his talk, Giovanni announced a new criterion for the ergodicity of the billiard flow on polygons (and, more generally, the geodesic flow on a flat surface with conical singularities) with potential applications to a whole class (of full measure) of Diophantine polygons.

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Posted by: matheuscmss | November 11, 2014

On the speed of ergodicity of horocycle maps

Last Thursday (November 6, 2014), Giovanni Forni gave a 1-hour talk at Orsay about his joint work with Livio Flaminio and James Tanis on the speed of ergodicity of horocycle maps.

In this blog post, I will transcript my notes for Giovanni’s talk. Of course, all mistakes in this post are my entire responsibility. Also, I apologize in advance for any wrong statements in what follows: indeed, I arrived at the seminar room about 10 minutes after Giovanni’s talk had started; furthermore, since the seminar room was crowded (about 30 to 40 mathematicians were attending the talk), I was forced to sit in the back of the room and consequently sometimes I could not properly hear Giovanni’s explanations.

The main actor in Giovanni’s talk was the classical horocycle flow {h_t}. By definition, {h_t} is the flow induced by the action of the {1}-parameter subgroup {\left( \begin{array}{cc} 1 & t \\ 0 & 1 \end{array}\right)} on the unit cotangent bundle {SL(2,\mathbb{R})/\Gamma=:M} of a hyperbolic surface {\mathbb{H}/\Gamma} of finite area (i.e., {\Gamma} is a lattice of {SL(2,\mathbb{R})}).

The optimal speed of ergodicity (rate of convergence of Birkhoff averages) for classical horocycle flows was the subject of several papers in the literature of Dynamical Systems: for example, after the works of Zagier, Sarnak, Burger, Ratner, Flaminio-Forni, Strömbergsson, etc., we know that the rate of ergodicity is intimately related to the eigenvalues of the Laplacian (“size of the spectral gap”) of the corresponding hyperbolic surface (and, furthermore, this is related to the Riemann hypothesis in the case {\Gamma= SL(2,\mathbb{Z})}).

The bulk of Giovanni’s talk was the discussion of the analog problem for horocycle maps, that is, the question of determining the optimal {\alpha>0} such that the iterates of the time {T} map of the horocycle flow {h_t} verify

\displaystyle \left|\sum\limits_{n=0}^{N-1} f(h_{nT}(x)) - N \int f d \textrm{vol}_M\right|\leq C_T N^{1-\alpha}

The basic motivations behind this question are potential applications to “sparse equidistribution problems” (some of them coming from Number Theory) such as:

  • The following particular case of Sarnak’s conjecture on the randomness of Möbius function: for all {x\in M=SL(2,\mathbb{R})/SL(2,\mathbb{Z})} and {f\in C^0(M)}, one has

    \displaystyle \frac{1}{\#\{p\leq P: p \textrm{ is prime}\}}\sum\limits_{\stackrel{p\leq P,}{p \textrm{ prime}}} f(h_p(x)) \stackrel{P\rightarrow\infty}{\longrightarrow} \int f d\textrm{vol}_M.

    In other words, the non-conventional ergodic averages of the horocycle flow along prime numbers at every point converge to the spatial average.

  • N. Shah’s conjecture: for each {\gamma>0}, one has

    \displaystyle \frac{1}{N}\sum\limits_{n=0}^{N-1} f(h_{n^{1+\gamma}}(x)) \rightarrow \int f d\textrm{vol}_M

    for all {x\in M} and {f\in C^0(M)} whenever {M=SL(2,\mathbb{R})/\Gamma} with {\Gamma} cocompact (i.e., the hyperbolic surface {\mathbb{H}/\Gamma} is compact). In other terms, the non-conventional ergodic averages of the horocycle flow along a polynomial sequence of times of the form {n^{1+\gamma}}, {\gamma>0}, at every point converge to the spatial average.

Also, Giovanni expects that the tools developed to obtain an estimate of the form {\left|\sum\limits_{n=0}^{N-1} f(h_{nT}(x)) - N \int f d \textrm{vol}_M\right|\leq C_T N^{1-\alpha}} could help in deriving quantitative versions of Ratner’s equidistribution results in more general contexts than the classical horocycles flows.

Before stating some of the main results of Flaminio, Forni and Tanis, let us just mention that:

  • Sarnak and Ubis gave in 2011 the following evidence towards the particular case of Sarnak’s conjecture stated above: every weak-{\ast} limit of the sequence of probability measures

    \displaystyle \frac{1}{\#\{p\leq P: p \textrm{ is prime}\}}\sum\limits_{\stackrel{p\leq P,}{p \textrm{ prime}}} \delta_{h_p(x)}

    converges to an absolutely continuous measure (with respect to {\textrm{vol}_M}) whose density is bounded by {10}. (Here, {\delta_z} is the usual Dirac mass at {z});

  • Very roughly speaking, an evidence in favor of Shah’s conjecture for {\gamma>0} very small is the fact that {n^{1+\gamma}} behaves like a linear function {n^{1+\gamma}\sim N^{\gamma}n} (with a mildly large factor {N^{\gamma}}) for {n\sim N}, so that Shah’s conjecture should not be very far from the corresponding statement of equidistribution for linear sequences of times {nT}. As it turns out, Flaminio, Forni and Tanis were able to convert this heuristic argument in a proof of Shah’s conjecture for {\gamma>0} very small: indeed, they are confident that Shah’s conjecture is settled for {0<\gamma<1/20} and they hope to push their methods to get the same results for {0<\gamma<1/11}. Here, a key ingredient is Theorem 1 below where Flaminio, Forni and Tanis establish a precise control on the quantity {\left|\sum\limits_{n=0}^{N-1} f(h_{nT}(x)) - N \int f d \textrm{vol}_M\right|}.

After this brief introduction to horocycle maps, we are ready to state the main result of this post:

Theorem 1 (Flaminio-Forni-Tanis) Let {\Gamma} be a cocompact subgroup of {SL(2,\mathbb{R})} and fix {T>0}. For all {x\in M = SL(2,\mathbb{R})/\Gamma} and {f\in C^{\infty}(M)}, one has

\displaystyle \left|\sum\limits_{n=0}^{N-1} f(h_{nT}(x)) - N\int f d\textrm{vol}_M\right|\leq C N^{(1+\nu_0)/2} \|f\| + C_T N^{5/6} (\log N)^{1/2} \|f\| \ \ \ \ \ (1)

where {\mu_0>0} is the smallest eigenvalue of the Laplacian {\Delta} on the hyperbolic surface {\mathbb{H}/\Gamma}, {\nu_0} is the following quantity (related to the so-called spectral gap):

\displaystyle \nu_0=\left\{\begin{array}{cl}\sqrt{1-4\mu_0} & \textrm{if } \mu_0\leq 1/4 \\ 0 & \textrm{otherwise}\end{array}\right.,

{\|f\|} is an adequate Sobolev norm {H^s} (say, {s=12}, i.e., {\|f\|} depends on the first twelve derivatives of {f}), {C} is an “universal” constant (depending on {\Gamma} only) and {C_T} is a constant depending on {T} and {\Gamma}.

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Posted by: matheuscmss | October 27, 2014

A family of maps preserving the measure of ZxT

Last 15th October 2014, the “flat seminar” coorganized by Anton Zorich, Jean-Christophe Yoccoz and myself restarted in a new format: instead of one talk per week, we shifted to one talk per month.

The first talk of this seminar in this new format was given by Alba Málaga, and the next two talks (on next November 12th and December 10th) will be given by Giovanni Forni (on the ergodicity for billiards in irrational polygons) and James Tanis (on equidistribution for horocycle maps): the details can be found here.

In this blog post, we will discuss Alba’s talk about some of the results in her PhD thesis (under the supervision of J.-C. Yoccoz) concerning a family of maps preserving the measure of {\mathbb{Z}\times\mathbb{T}} (as hinted by the title of this post). Of course, any mistakes/errors in what follows are my entire responsibility.

In her PhD thesis, Alba studies the following family of dynamical systems (“cylinder flows”).

The phase space is {\mathbb{Z}\times\mathbb{T}} where {\mathbb{T}=\mathbb{R}/\mathbb{Z}} is the unit circle. We call {\{n\}\times\mathbb{T}} the circle of level {n\in\mathbb{Z}} in the phase space.

The parameter space is {\mathbb{T}^{\mathbb{Z}}:=\{\underline{\alpha}=(\dots, \alpha_{-1}, \alpha_0, \alpha_1,\dots): \alpha_n\in\mathbb{T} \,\,\,\, \forall \, n\in\mathbb{Z}\}}.

Given a parameter {\underline{\alpha}\in \mathbb{T}^{\mathbb{Z}}}, we can define a transformation {F_{\underline{\alpha}}} of the phase space {\mathbb{Z}\times\mathbb{T}} by rotating the elements {(n,x)} of the circle of level {n} by {\alpha_n}, and then by putting them at the level {n+1} (one level up) or {n-1} (one level down) depending on whether they fall in the first or second half of the circle of level {n}. In other terms,

\displaystyle F_{\underline{\alpha}}(n,x):=\left\{\begin{array}{cl} (n+1, R_{\alpha_n}x) & \textrm{if } 0 < R_{\alpha_n}x < 1/2 \\ (n-1, R_{\alpha_n}x) & \textrm{if } -1/2 < R_{\alpha_n}x < 0 \end{array}\right.

where {R_{\alpha}x:=x+\alpha} is the rotation by {\alpha} on the unit circle {\mathbb{T}}.

Note that we have left {F_{\underline{\alpha}}} undefined at the points {(n,x)} such that {R_{\alpha_n}x=0} or {R_{\alpha_n}x=1/2}. Of course, one can complete the definition of {F_{\underline{\alpha}}} by sending each of the points in this countable family to a level up or down in an arbitrary way. However, we prefer not do so because this countable family of points will play no role in our discuss of typical orbits of {F_{\underline{\alpha}}}. Instead, we will think of the set {\textrm{sing}(F_{\underline{\alpha}})} of points where {F_{\underline{\alpha}}} is undefined as a (very mild) singular set.

Alba’s initial motivation for studying this family comes from billiards in irrational polygons. Indeed, our current knowledge of the dynamics of billiard maps on irrational polygons (i.e., polygons whose angles are not all rational multiples of {\pi}) is very poor, and, as Alba explained very well in her talk (with the aid of computer-made figures), she has a good heuristic argument suggesting that the billiard map on an irrational lozenge obtained by small perturbation of an unit square can be thought as a small perturbation of some members of the family {F_{\underline{\alpha}}}. However, we will not pursue further this direction today and we will focus exclusively on the features of {F_{\underline{\alpha}}} from now on.

It is an easy exercise to check that, for any parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}}, the corresponding dynamical system {F_{\underline{\alpha}}} preserves the infinite product measure {\mu=\nu\times \textrm{Leb}}, where {\nu} is the counting measure on {\mathbb{Z}} and {\textrm{Leb}} is the Lebesgue measure on {\mathbb{T}}.

In this setting, Alba’s thesis is concerned with the dynamics of {F_{\underline{\alpha}}} for a typical parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (in both Baire-category and measure-theoretical senses).

Before stating some of Alba’s results, let us quickly discuss the dynamical behavior of {F_{\underline{\alpha}}} for some particular choices of the parameter {\underline{\alpha}\in \mathbb{T}^{\mathbb{Z}}}.

Example 1 Consider the constant sequence {(1/2)_{\infty}:=(\dots, 1/2, 1/2, 1/2,\dots)}. By definition, {F_{(1/2)_{\infty}}} acts by a translation by {1/2} on the {x}-coordinate of all points {(n,x)} of the phase space. In particular, the second iterate {F_{(1/2)_{\infty}}^2(n,x)} of any point {(n,x)} has the form {F_{(1/2)_{\infty}}^2(n,x) = (c_{1/2}(n,x),x)} where {c_{1/2}(n,x)\in\mathbb{Z}}. Furthermore, the function {c_{1/2}(n,x)} is not difficult to compute: since {0<x<1/2 \iff 1/2<x+1/2<1}, we see that if {0<x<1/2}, resp. {1/2<x<1}, then {F_{(1/2)_{\infty}}(n,x) = (n-1,x+1/2)} resp. {(n+1,x+1/2)} and, hence, {F_{(1/2)_{\infty}}^2(n,x)=(n,x)}. In other words, {c_{1/2}(n,x)=n} for all {(n,x)}, and, thus, {F_{(1/2)_{\infty}}} is a periodic transformation (of period two).

Example 2 Consider the constant sequence {(1/3)_{\infty}:=(\dots, 1/3, 1/3, 1/3, \dots)}. Similarly to the previous example, {F_{(1/3)_{\infty}}} acts periodically (with period {3}) on the {x}-coordinate in the sense that {F_{(1/3)_{\infty}}^3(n,x)=(c_{1/3}(n,x),x)} where {c_{1/3}(n,x)\in\mathbb{Z}}. Again, the function {c_{1/3}(n,x)} is not difficult to compute: by dividing the unit circle {\mathbb{T}} into the six intervals {I_j = \left[ \frac{j}{6}, \frac{j+1}{6} \right]}, {j=0,\dots, 5}, one can easily check that

\displaystyle c_{1/3}(n,x)=\left\{\begin{array}{cl} n+1 & \textrm{if } x\in I_j \textrm{ with } j \textrm{ even } \\ n-1 & \textrm{if } x\in I_j \textrm{ with } j \textrm{ odd } \end{array} \right.

In particular, we see that {F_{(1/3)_{\infty}}^3} systematically moves the copy {\{n\}\times I_j} of an interval {I_j} with {j} even, resp. odd, at the circle of level {n} to the corresponding copy {\{n+1\}\times I_j}, resp. {\{n-1\}\times I_j}, of the interval {I_j} at level {n+1}, resp. {n-1}. In other terms, {F_{(1/3)_{\infty}}} has wandering domains (i.e., domains which are disjoint from all its non-trivial iterates under the map) of positive {\mu}-measure and, hence, {F_{(1/3)_{\infty}}} is not conservative in the sense that it does not satisfy Poincaré’s recurrence theorem with respect to the infinite invariant measure {\mu}: for example, for each {k\in\mathbb{N}}, {F_{(1/3)_{\infty}}^{3k}} sends the subset {\{0\}\times I_0} of {\mu}-measure {1/6} always “upstairs” to its copy {\{k\}\times I_0} at the {k}-th level, so that the orbits of points in {\{0\}\times I_0} escape to {+\infty} (one of the “ends”) in the phase space {\mathbb{Z}\times\mathbb{T}}.

Remark 1 The reader can easily generalize the previous two examples to obtain that the transformation {F_{(p/q)_{\infty}}} associated to the constant sequence {(p/q)_{\infty} = (\dots, p/q, p/q, p/q,\dots)} with {p/q\in\mathbb{Q}} (a rational number written in lowest terms) is periodic or it has wandering domains of positive measure depending on whether the denominator {q\in\mathbb{N}} is even or odd.

Example 3 By a theorem of Conze and Keane, the transformation {F_{(\alpha)_{\infty}}} associated to a constant sequence {(\alpha)_{\infty} = (\dots, \alpha, \alpha, \alpha, \dots)} with {\alpha\in\mathbb{R}-\mathbb{Q}} is ergodic (but not minimal).

Today, we will give sketches of the proofs of the following two results:

Theorem 1 (Málaga) For almost all parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (with respect to the standard product Lebesgue measure), the transformation {F_{\underline{\alpha}}} is conservative, i.e., {F_{\underline{\alpha}}} has no wandering domains of positive {\mu}-measure.

Theorem 2 (Málaga) For a Baire-generic parameter {\underline{\alpha}\in\mathbb{T}^{\mathbb{Z}}} (with respect to the standard product topology), the transformation {F_{\underline{\alpha}}} is conservative, ergodic, and minimal.

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Posted by: matheuscmss | August 31, 2014

Dynamics of the Weil-Petersson flow: rates of mixing

For the last installment of this series, our goal is to discuss the rates of mixing of the Weil-Petersson (WP) geodesic flow on the unit tangent bundle {T^1\mathcal{M}_{g,n}} of the moduli space {\mathcal{M}_{g,n}} of Riemann surfaces of genus {g\geq 0} with {n\geq 0} punctures for {3g-3+n\geq 1}.

However, before entering into the mathematical discussion strictly speaking, let me take the opportunity to dedicate this blog post to the memory of two Russian mathematicians who passed away earlier this month: Dmitri Anosov and Nikolai Chernov. Among their several well-known contributions in Dynamical Systems, we can quote:

Of course, the list of contributions of Anosov and Chernov to Dynamical Systems is vast: each of them wrote more than 90 research articles and books about the features of systems with some hyperbolicity (such as geodesic flows on negatively curved manifolds and chaotic billiards) among other topics.

In particular, it is out of the scope of this post to provide detailed descriptions of the works of these two very influential dynamicists.

On the other hand, as a form of “small compensation”, let me say that the second section of this post (about rates of the WP flow on the modular surface) briefly discusses some of the ideas advanced by these two mathematicians.

Concerning the rates of mixing of the WP flow, let us recall that, by Burns-Masur-Wilkinson theorem (cf. Theorem 1 in the first post of this series), the WP flow {\varphi_t} on {T^1\mathcal{M}_{g,n}} is mixing with respect to the Liouville measure {\mu} whenever {3g-3+n\geq 1}.

By definition of the mixing property, this means that the correlation function {C_t(f,g):=\int f \cdot g\circ\varphi_t d\mu - \left(\int f d\mu\right)\left(\int g d\mu\right)} converges to {0} as {t\rightarrow\infty} for any given {L^2}-integrable observables {f} and {g}. (See, e.g., the section “{L^2} formulation” in this Wikipedia article about the mixing property.)

Given this scenario, it is natural to ask how fast the correlation function {C_t(f,g)} converges to zero. In general, the correlation function {C_t(f,g)} can decay to {0} (as a function of {t\rightarrow\infty}) in a very slow way depending on the choice of the observables (see, e.g., this blog post of Climenhaga for some concrete examples). Nevertheless, it is often the case (for mixing flows with some hyperbolicity) that the correlation function {C_t(f,g)} decays to {0} with a definite (e.g., polynomial, exponential, etc.) speed when restricting the observables to appropriate spaces of “reasonably smooth” functions.

In other words, given a mixing flow (with some hyperbolicity), it is usually possible to choose appropriate functional (e.g., Hölder, {C^r}, Sobolev, etc.) spaces {X} and {Y} such that

  • {|C_t(f,g)|\leq C\|f\|_{X} \|g\|_{Y} t^{-n}} for some constants {C>0}, {n\in\mathbb{N}} and for all {t\geq 1} (polynomial decay),
  • or {|C_t(f,g)|\leq C\|f\|_{X} \|g\|_{Y} e^{-ct}} for some constants {C>0}, {c>0} and for all {t\geq 1} (exponential decay).

Evidently, the “precise” rate of mixing of the flow (i.e., the sharp values of the constants {C>0}, {n\in\mathbb{N}} and/or {c>0} above) depend on the choice of the functional spaces {X} and {Y} (as they might change if we replace {C^1} observables by {C^2} observables say). On the other hand, the qualitative speed of decay of {C_t(f,g)}, that is, the fact that {C_t(f,g)} decays polynomially or exponentially as {t\rightarrow\infty} whenever {f} and {g} are “reasonably smooth”, remains unchanged if we select {X} and {Y} from a well-behaved scale of functional (like {C^r} spaces, {r\in\mathbb{N}}, or {H^s} spaces, {s>0}). In particular, this partly explains why in the Dynamical Systems literature one simply says that a given mixing flow {\varphi_t} has “polynomial decay” or “exponential decay”: usually we are interested in the qualitative behavior of the correlation function for reasonably smooth observables, but the particular choice of functional spaces {X} and {Y} is normally treated as a “technical detail”.

After this brief description of the notion of rate of mixing (speed of decay of correlation functions), we are ready to state the main result of this post.

Theorem 1 (Burns-Masur-M.-Wilkinson) The rate of mixing of the WP flow {\varphi_t} on {T^1\mathcal{M}_{g,n}} is:

  • at most polynomial when {3g-3+n>1};
  • rapid (faster than any polynomial) when {3g-3+n=1}.

Remark 1 This result was announced as Theorem 2 in the first post of this series and also in this preprint here. Since then, Burns, Masur, Wilkinson and myself found some evidence indicating that the Weil-Petersson geodesic flow on {T^1\mathcal{M}_{g,n}} is actually exponentially mixing when {3g-3+n=1}. The details will hopefully appear in the forthcoming paper (currently still in preparation).

Remark 2 An open problem left by Theorem 1 is to determine the rate of mixing of the WP flow on {T^1\mathcal{M}_{g,n}} for {3g-3+n>1}. Indeed, while this theorem provides a polynomial upper bound for the rate of mixing in this setting, it does not rule out the possibility that the actual rate of mixing of the WP flow is sub-polynomial (even for reasonably smooth observables). Heuristically speaking, we believe that the sectional curvatures of the WP metric control the time spend by WP geodesics near the boundary of {\overline{\mathcal{M}}_{g,n}}. In particular, it seems that the problem of determining the rate of mixing of the WP flow (when {3g-3+n>1}) is somewhat related to the issue of finding suitable (polynomial?) bounds for how close to zero the sectional curvatures of the WP metric can be (in terms of the distance to the boundary of {\overline{\mathcal{M}}_{g,n}}). Unfortunately, the best available bounds for the sectional curvatures of the WP metric (due to Wolpert) do not rule out the possibility that some of these quantities get extremely close to zero (see Remark 4 of this post here).

The difference in the rates of mixing of the WP flow on {T^1\mathcal{M}_{g,n}} when {3g-3+n>1} or {3g-3+n=1} in Theorem 1 reflects the following simple (yet important) feature of the WP metric near the boundary of the Deligne-Mumford compactification of {\mathcal{M}_{g,n}}.

In the case {3g-3+n=1}, e.g., {g=1=n}, the moduli space {\mathcal{M}_{1,1}\simeq\mathbb{H}/PSL(2,\mathbb{Z})} equipped with the WP metric looks like the surface of revolution of the profile {\{v=u^3: 0 < u \leq 1\}} near the cusp at infinity (see Remark 6 of this post here). In particular, even though a {\varepsilon}-neighborhood of the cusp is “polynomially large” (with area {\sim \varepsilon^4}), the Gaussian curvature approaches only {-\infty} near the cusp and, as it turns out, this strong negative curvature near the cusp makes that all geodesic not pointing directly towards the cusp actually come back to the compact part in bounded (say {\leq 1}) time. In other words, the excursions of infinite WP geodesics on {\mathcal{M}_{1,1}} near the cusp are so quick that the WP flow on {T^1\mathcal{M}_{1,1}} is “close” to a classical Anosov geodesic flow on negatively curved compact surface. In particular, it is not entirely surprising that the WP flow on {T^1\mathcal{M}_{1,1}} is rapid.

On the other hand, in the case {3g-3+n>1}, the WP metric on {\mathcal{M}_{g,n}} has some sectional curvatures close to zero near the boundary of the Deligne-Mumford compactification {\overline{\mathcal{M}}_{g,n}} of {\mathcal{M}_{g,n}} (see Theorem 3 and Remark 5 of this post here). By exploiting this feature of the WP metric on {\mathcal{M}_{g,n}} for {3g-3+n>1} (that has no counterpart for {\mathcal{M}_{1,1}} or {\mathcal{M}_{0,4}}), we will build a non-neglegible set of WP geodesics spending a long time near the boundary of {\overline{\mathcal{M}}_{g,n}} before eventually getting into the compact part. In this way, we will deduce that the WP flow on {\mathcal{M}_{g,n}} takes a fair (polynomial) amount of time to mix certain parts of the boundary of {\overline{\mathcal{M}}_{g,n}} with fixed compact subsets of {\mathcal{M}_{g,n}}.

In the remainder of this post, we will give some details of the proof of Theorem 1. In the next section, we give a fairly complete proof (assuming the results in this previous post, of course) of the polynomial upper bound on the rate of mixing of the WP flow on {T^1\mathcal{M}_{g,n}} when {3g-3+n>1}. After that, in the final section, we provide a sketch of the proof of the rapid mixing property of the WP flow on {T^1\mathcal{M}_{1,1}}. In fact, we decided (for pedagogical reasons) to explain some key points of the rapid mixing property only in the toy model case of a negatively curved surface with one cusp corresponding exactly to a surface of revolution of a profile {\{v=u^r\}}, {r\geq 2}. In this way, since the WP metric near the cusp of {\mathcal{M}_{1,1}\simeq \mathbb{H}/PSL(2,\mathbb{Z})} can be thought as a “perturbation” of the surface of revolution of {\{v=u^3\}} (thanks to Wolpert’s asymptotic formulas), the reader hopefully will get a flavor of the main ideas behind the proof of rapid mixing of the WP flow on {\mathcal{M}_{1,1}} without getting into the (somewhat boring) technical details needed to check that the arguments used in the toy model case are “sufficiently robust” so that they can be “carried over” to the “perturbative setting” of the WP flow on {T^1\mathcal{M}_{1,1}}.

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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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