Posted by: matheuscmss | March 15, 2015

First Bourbaki seminar of 2015 (III): Ambrosio’s talk

For the third (and last) installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss a talk that brought me back some good memories from the time I did my PhD at IMPA when I took a course on Geometric Measure Theory (taught by Hermano Frid) based on the books of Evans-Gariepy (for the introductory part of the course) and Guisti (for the core part of the course).

In fact, our goal today is to revisit Luigi Ambrosio’s Bourbaki seminar talk entitled “The regularity theory of area-minimizing integral currents (after Almgren-DeLellis-Spadaro)”. Here, besides the original works of Almgren and DeLellis-Spadaro (in these five papers here), the main references are the video of Ambrosio’s talk and his lecture notes (both in English).

Disclaimer. As usual, all errors and mistakes are my entire responsibility.

1. Introduction

This post is centered around solutions to the so-called Plateau’s problem.

A formulation of Plateau’s problem in dimension {m} and codimension {n} is the following. Given a {(m+n)}-dimensional Riemannian manifold and a {(m-1)}-dimensional compact embedded oriented submanifold {\Gamma\subset M} (without boundary), find a {m}-dimensional embedded oriented submanifold {\Sigma\subset M} with boundary {\partial\Sigma=\Gamma} such that

\displaystyle \textrm{vol}_m(\Sigma)\leq \textrm{vol}_m(\widetilde{\Sigma})

for all oriented {m}-dimensional submanifold {\widetilde{\Sigma}} with {\partial\widetilde{\Sigma} = \Gamma}. (Here, {\textrm{vol}_m(A)} denotes the {m}-dimensional volume of {A}).

This formulation of Plateau’s problem allows for several variants. Moreover, the solution to Plateau’s problem is very sensitive on the precise mathematical formulation of the problem (and, in particular, on the dimension {m} and codimension {n}).

Example 1 The works of Douglas and Radó (based on the conformal parametrization method and some compactness arguments) provided a solution to (a formulation of) Plateau’s problem when {m=2} and the boundary {\Gamma} is circular (i.e., {\Gamma} is parametrized by the round circle {S^1=\{ (x,y)\in\mathbb{R}^2: x^2+y^2=1 \}}). Unfortunately, the techniques employed by Douglas and Radó do not work for arbitrary dimension {m} and codimension {n}.

Example 2 The following example gives an idea on the difficulties that one might found while trying to solve Plateau’s problem (in the formulation given above).Let us consider the case {m=n=2}, {\Gamma\subset\mathbb{R}^4\simeq \mathbb{C}^2} defined as

\displaystyle \Gamma:=\{(\zeta^2,\zeta^3)\in\mathbb{C}^2: |\zeta|=1\}

The singular immersed disk

\displaystyle D:=\{(z, w)\in\mathbb{C}^2: z^3=w^2, |z|\leq 1\}

satisfies {\partial D=\Gamma} and the so-called calibration method can be applied to prove that

\displaystyle \mathcal{H}^2(D)<\mathcal{H}^2(\Sigma)

for all smooth oriented {2}-dimensional submanifold {\Sigma\subset\mathbb{R}^4} with {\partial\Sigma = \Gamma}. (Here, {\mathcal{H}^2} stands for the {2}-dimensional Hausdorff measure on {\mathbb{R}^4}.)

The example above motivates the introduction of weak solutions (including immersed submanifolds) to Plateau’s problem, so that the main question becomes the existence and regularity of weak solutions.

This point of view was adopted by several authors: for example, De Giorgi studied the notion of sets of finite perimeter when {n=1}, and Federer and Fleming introduced the notion of currents.

Remark 1 The “PDE counterpart” of this point of view is the study of existence and regularity of weak solutions of PDEs in Sobolev spaces.

In arbitrary dimensions and codimensions, De Giorgi’s regularity theory provides weak solutions to Plateau’s problem which are smooth on open and dense subsets.

As it was pointed out by Federer, this is not a satisfactory regularity statement. Indeed, De Giorgi’s regularity theory allows (in principle) weak solutions to Plateau’s problem whose singular set (i.e., the subset of points where the weak solution is not smooth) could be “large” in the sense that its ({m}-dimensional) Hausdorff measure could be positive.

In this context, Almgren wrote two preprints which together had more than 1000 pages (published in posthumous way here) where it was shown that the singular set of weak solutions to Plateau’s problem has codimension {2} at least:

Theorem 1 (Almgren) Let {T} be an integral rectifiable {m}-dimensional current in a {(m+n)}-dimensional {C^5} Riemannian manifold {M^{m+n}}.If {T} is area-minimizing (i.e., a weak solution to Plateau’s problem), then there exists a closed subset {\textrm{sing}(T)} of {T} such that:

  • {\textrm{sing}(T)} has codimension {\geq 2}: the Hausdorff dimension of {\textrm{sing}(T)} is {\leq (m-2)}, and
  • {\textrm{sing}(T)} is the singular set of {T}: the subset {T\cap (M-(\textrm{supp}(\partial T)\cup\textrm{sing}(T))} is induced by a smooth oriented {m}-dimensional submanifold of {M}.

We will explain the notion of area-minimizing integral rectifiable currents (appearing in the statement above) in a moment. For now, let us just make some historical remarks. Ambrosio has the impression that some parts of Almgren’s work were not completely reviewed, even though several experts have used some of the ideas and techniques introduced by Almgren. For this reason, the simplifications (of about {1/3}) and, more importantly, improvements of Almgren’s work obtained by DeLellis-Spadaro in a series of five articles (quoted in the very beginning of the pots) were very much appreciated by the experts of this field.

Closing this introduction, let us present the plan for the remaining sections of this post:

  • the next section reviews the construction of solutions to the generalized Plateau problem via Federer-Fleming compactness theorem for integral rectifiable currents;
  • then, in the subsequent section, we revisit some aspects of the regularity theory of weak solutions to Plateau’s problem in codimension {n=1}; in particular, we will see in this setting a stronger version of Theorem 1;
  • after that, in the last section of this post, we make some comments on the works of Almgren and DeLellis-Spadaro on the regularity theory for Plateau’s problem in codimension {n\geq 2}; in particular, we will sketch the proof of Theorem 1 above.

Remark 2 For the sake of exposition, from now on we will restrict ourselves to the study of Plateau’s problem in the case of Euclidean ambient spaces, i.e., {M^{m+n} = \mathbb{R}^{m+n}}. In fact, this is not a great loss in generality because the statements and proofs in the Euclidean setting {M^{m+n} = \mathbb{R}^{m+n}} can be adapted to arbitrary Riemannian manifolds {M^{m+n}} with almost no extra effort.

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Posted by: matheuscmss | March 7, 2015

Journée Surfaces Plates

Luca Marchese and I are organizing a one-day event called “Journée Surfaces Plates” at LAGA, Université Paris 13 next May 20th, 2015.

This event will consist into three talks by Giovanni Forni, Jean-Christophe Yoccoz and Anton Zorich, and a tentative schedule is available here. Also, it is likely that these talks will be recorded, and, in this case, I plan to update this (very short) post by providing a link for the eventual videos of these lectures.

Please note that any interested person can attend this event (as there are no inscription fees).  On the other hand, since our budget is very limited, unfortunately Luca and I  can not offer any sort of financial help (with local and/or travel expenses) to potential participants. In particular, we would ask you to use your own grants to support your eventual participation in “Journée Surfaces Plates”.

Posted by: matheuscmss | February 28, 2015

First Bourbaki seminar of 2015 (II): Carron’s talk

For the second installment of this series of posts (which started here) on the first Bourbaki seminar of 2015, we will discuss Gilles Carron talk entitled “New utilisation of the maximum principles in Geometry (after B. Andrews, S. Brendle, J. Clutterbuck)”. Here, besides the original works of Andrews-Clutterbuck and Brendle (quoted below), the main references are the video of Carron’s talk and his lecture notes (both in French).

Disclaimer. All errors, mistakes or misattributions are my entire responsibility.

1. Introduction

Given a Riemannian {n}-dimensional manifold {M}, one can often study its Geometry by analyzing adequate smooth real functions {f} on {M} (such as scalar curvature). One of the techniques used to get some information about {f} is the following observation (“baby maximum principle”): if {f} has a local maximum at a point {p}, then we dispose of

  • a first order information: the gradient of {f} at {p} vanishes; and
  • a second order information: the Hessian of {f} at {p} has a sign (namely, it is negative definite).

In order to extract more information from this technique, one can appeal to the so-called doubling of variables method: instead of studying {f}, one investigates the local maxima of a “well-chosen” function {g} on the double of variables (e.g., {g:M\times M\rightarrow\mathbb{R}}). In this way, we have new constraints because the gradient and Hessian of {g} depend on more variables than those of {f}.

This idea of doubling the variables goes back to Kruzkov who used it to estimate the modulus of continuity of the derivative of solutions of a non-linear parabolic PDE (in one space dimension). In this post we shall see how this idea was ingeniously employed by Andrews and Clutterbuck (2011) and Brendle (2013) in two recent important works.

We start with the statement of Andrews-Clutterbuck theorem:

Theorem 1 (Andrews-Clutterbuck) Let {\Omega\subset\mathbb{R}^n} be a convex domain of diameter {D}. Consider the Schrödinger operator {-\Delta+V} where {\Delta=\sum\limits_{i=1}^n\frac{\partial^2}{\partial^2 x_i}} is the Laplacian operator and {V} is the operator induced by the multiplication by a convex function {V:\Omega\rightarrow\mathbb{R}}.Recall that the spectrum of {-\Delta+V} with respect to Dirichlet condition on the boundary {\partial\Omega} consists of a discrete set of eigenvalues of the form: {\lambda_1<\lambda_2\leq \dots}

In this setting, the fundamental gap {\lambda_2-\lambda_1} of {-\Delta+V} is bounded from below by

\displaystyle \lambda_2 - \lambda_1 \geq 3\frac{\pi^2}{D^2}

Remark 1 This theorem is sharp: {\lambda_2-\lambda_1=3\frac{\pi^2}{D^2}} when {\Omega=(-D/2, D/2)\subset\mathbb{R}} and {V\equiv 0} (by Fourier analysis). In other terms, Andrews-Clutterbuck theorem is an optimal comparison theorem between the fundamental gap of general Schrödinger operators with the one-dimensional case.

Next, we state Brendle’s theorem:

Theorem 2 (Brendle) A minimal torus inside the round sphere {S^3=\{(x_1,\dots, x_4\in\mathbb{R}^4: x_1^2+\dots+x_4^2=1\}} is isometric to Clifford torus {\mathbb{T}=\{(x_1,\dots,x_4)\in\mathbb{R}^4: x_1^2+x_2^2 = x_3^2+x_4^2 = 1/2\}}.

The sketches of proof of these results are presented in the next two Sections. For now, let us close this introductory section by explaining some of the motivations of these theorems.

1.1. The context of Andrews-Clutterbuck theorem

The interest of the fundamental gap {\gamma=\lambda_2-\lambda_1} comes from the fact that it helps in the description of the long-term behavior of non-negative non-trivial solutions of the heat equation

\displaystyle \frac{\partial}{\partial t} u(t,x) = \Delta u(t,x) - V(x)u(t,x), \quad (t,x)\in [0,\infty)\times \Omega

with {u\equiv 0} on {\partial \Omega}. More precisely, one has that

\displaystyle u(t,x) = c \exp(\lambda_1t) f_1(x) (1+O(\exp(-\gamma t))


  • {c} is an adequate constant,
  • {f_1} is the ground state of {-\Delta+V}, i.e., {-\Delta f_1+V f_1=\lambda_1 f_1}, {f_1>0} on {\textrm{int}(\Omega)}, {f_1=0} on {\partial\Omega} and {f_1} is normalized so that {\int_{\Omega} f_1^2=1}, and
  • {O(\exp(-\gamma t))} denotes (as usual) a quantity bounded from above by {C\exp(-\gamma t)} for some constant {C>0} and all {t\geq 0}.

The theorem of Andrews-Clutterbuck answers positively a conjecture of Yau and Ashbaugh-Benguria. This conjecture was based on a series of works in Mathematics and Physics: from the mathematical side, van den Berg observed during his study of the behavior of spectral functions in big convex domains (modeling Bose-Einstein condensation) that {\lambda_2-\lambda_1\geq 3\pi^2/D^2} for the free Laplacian ({V\equiv 0}) on several convex domains. After that, Singer-Wong-Yau-Yau proved that

\displaystyle \lambda_2-\lambda_1\geq \frac{1}{4}\left(\frac{\pi^2}{D^2}\right)

and Yu-Zhong improved this result by showing that

\displaystyle \lambda_2-\lambda_1\geq \frac{\pi^2}{D^2}

Furthermore, some particular cases of Andrews-Clutterbuck were previously known: for instance, Lavine proved the one-dimensional case {\Omega\subset\mathbb{R}}, and other authors studied the cases of convex domains with some (axial and/or rotational) symmetries in higher dimensions.

1.2. The context of Brendle theorem

The theorem of Brendle answers affirmatively a Lawson’s conjecture.

Lawson arrived at this conjecture after proving (in this paper here) that every compact oriented surface {\Sigma} without boundary can be minimally embedded in {S^3}.

Remark 2 The analog of Lawson’s theorem is completely false in {\mathbb{R}^3}: using the maximum principle, one can show that there are no immersed compact minimal surfaces in {\mathbb{R}^3}.

Moreover, Lawson (in the same paper loc. cit.) showed that, if the genus of {\Sigma} is not prime, then {\Sigma} admits two non-isometric minimal embeddings in {S^3}.

On the other hand, Lawson’s construction in the case of genus {1} produces only the Clifford torus (up to isometries). Nevertheless, Lawson proved (in this paper here) that if {\Sigma\subset S^3} is a minimal torus, then there exists a diffeomorphism {F:S^3\rightarrow S^3} taking {\Sigma} to the Clifford torus {\mathbb{T}}: in other terms, there is no knotted minimal torus in {S^3}!

In this context, Lawson was led to conjecture that this diffeomorphism {F:S^3\rightarrow S^3} could be taken to be an isometry, an assertion that was confirmed by Brendle.

Read More…

Simion Filip, Giovanni Forni and I have just upload to ArXiv our paper Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups.

This article is motivated by Simion Filip’s recent work on the classification of possible monodromy groups for the Kontsevich-Zorich cocycle.

Very roughly speaking, the basic idea of this classification is the following. Consider the Kontsevich-Zorich cocycle on the Hodge bundle over the support of an ergodic {SL(2,\mathbb{R})}-invariant probability measure on (a connected component of) a stratum of the moduli spaces of translation surfaces. Recall that, in a certain sense, the Kontsevich-Zorich cocycle is a sort of “foliated monodromy representation” obtained by using the Gauss-Manin connection on the Hodge bundle while essentially moving only along {SL(2,\mathbb{R})}-orbits on moduli spaces of translation surfaces.

By extending a previous work of Martin Möller (for the Kontsevich-Zorich cocycle over Teichmüller curves), Simion Filip showed (in this paper here) that a version of the so-called Deligne’s semisimplicity theorem holds for the Kontsevich-Zorich cocycle: in plain terms, this means that the Kontsevich-Zorich cocycle can be completely decomposed into ({SL(2,\mathbb{R})}-)irreducible pieces, and, furthermore, each piece respects the Hodge structure coming from the Hodge bundle. In other terms, the Kontsevich-Zorich cocycle is always diagonalizable by blocks and its restriction to each block is related to a variation of Hodge structures of weight {1}.

The previous paragraph might seem abstract at first sight, but, as it turns out, it imposes geometrical constraints on the possible groups of matrices obtained by restriction of the Kontsevich-Zorich cocycle to an irreducible piece. More precisely, by exploiting the known tables (see § 3.2 of Filip’s paper) for monodromy representations coming from variations of Hodge structures of weight {1} over quasiprojective varieties, Simion Filip classified (up to compact and finite-index factors) the possible Zariski closures of the groups of matrices associated to restrictions of the Kontsevich-Zorich cocycle to an irreducible piece. In particular, there are at most five types of possible Zariski closures for blocks of the Kontsevich-Zorich cocycle (cf. Theorems 1.1 and 1.2 in Simion Filip’s paper):

Moreover, each of these items can be realized as an abstract variation of Hodge structures of weight {1} over abstract curves and/or Abelian varieties.

Here, it is worth to stress out that Filip’s classification of the possible blocks of the Kontsevich-Zorich cocycle comes from a general study of variations of Hodge structures of weight {1}. Thus, it is not clear whether all items above can actually be realized as a block of the Kontsevich-Zorich cocycle over the closure of some {SL(2,\mathbb{R})}-orbit in the moduli spaces of translations surfaces.

In fact, it was previously known in the literature that (all groups listed in) the items (i) and (ii) appear as blocks of the Kontsevich-Zorich cocycle (over closures of {S(2,\mathbb{R})}-orbits of translation surfaces given by certain cyclic cover constructions). On the other hand, it is not obvious that the other 3 items occur in the context of the Kontsevich-Zorich cocycle, and, indeed, this realizability question was explicitly posed by Simion Filip in Question 5.5 of his paper (see also § B.2 in Appendix B of this recent paper of Delecroix-Zorich).

In our paper, Filip, Forni and I give a partial answer to this question by showing that the case {SO^*(6)} of item (iv) is realizable as a block of the Kontsevich-Zorich cocycle.

Remark 1 Thanks to an exceptional isomorphism between the real Lie algebra {\mathfrak{so}^*(6)} in its standard representation and the second exterior power representation of the real Lie algebra {\mathfrak{su}(3,1)}, this also means that the case of {\wedge^2 SU(3,1)} of item (iii) is also realized.

Remark 2 We think that the examples constructed in this paper by Yoccoz, Zmiaikou and myself of regular origamis associated to the groups {SL(2,\mathbb{F}_p)} of Lie type might lead to the realizability of all groups {SO^*(2n)} in item (iv). In fact, what prevents Filip, Forni and I to show that this is the case is the absence of a systematic method to show that the natural candidates to blocks of the Kontsevich-Zorich cocycle over these examples are actually irreducible pieces.

In the remainder of this post, we will briefly explain our construction of an example of closed {SL(2,\mathbb{R})}-orbit such that the Kontsevich-Zorich cocycle over this orbit has a block where it acts through a Zariski dense subgroup of {SO^*(6)} (modulo compact and finite-index factors).

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Posted by: matheuscmss | February 22, 2015

First Bourbaki seminar of 2015 (I): Harari’s talk

About one month ago (on January 24, 2015), the first Bourbaki seminar of 2015 took place at Institut Henri Poincaré. As usual, this was an excellent opportunity to learn about recent advances in areas of Mathematics outside my field of expertise.

The first Bourbaki seminar of 2015 had the following four talks:

Today, I would like to discuss David Harari’s talk entitled “Zero cycles and rational points on fibrations in rationally connected varieties (after Harpaz and Wittenberg)”. Here, I will try to follow the first 38m50s of the video of Harari’s talk (in French) and sometimes his lecture notes (also in French). Of course, this goes without saying that any errors/mistakes are my full responsibility.

1. Introduction

One of the basic old problems in Number Theory is to determine whether a system of polynomial equations

\displaystyle P_i(x_1,\dots, x_n) = 0, \quad 1\leq i\leq r \ \ \ \ \ (1)

associated to homogeneous polynomials {P_i} with coefficients in a number field {k} has non-trivial solutions.

Equivalently, denoting by {X} the algebraic variety defined by the system (1), we want to know whether the set {X(k)} of points of {X} whose coordinates belong to {k} is not empty. In the literature, {X(k)} is called the set of {k}-rational points of {X}.

It is not easy to answer this problem in general. Nevertheless, we have the following necessary condition: if {X(k)\neq\emptyset}, then {X(k_v)\neq\emptyset} for all completion {k_v} of {k} with respect to a place of {k} (i.e., {v} is an equivalence class of absolute values). In other words, we have that {X(k)=\emptyset} whenever there is a local obstruction in the sense that {X(k_v)=\emptyset} for some place {v} of {k}.

This necessary condition based on local obstructions is helpful because it is often easy to verify algorithmically that {X(k_v)\neq\emptyset}. For example, when {k=\mathbb{Q}}, its completions {k_v} are either {k_v=\mathbb{Q}_p} (for the place of {p}-adic absolute values, {p\in\mathbb{N}} prime) or {k_v=\mathbb{R}=\mathbb{Q}_{\infty}} (for the “place at infinity”), and, in this situation, we can check that {X(\mathbb{Q}_p)\neq\emptyset} with the help of Hensel’s lemma ({p}-adic analog of Newton’s method).

It is known that this necessary condition is sufficient in certain special cases. For instance, the classical Hasse-Minkowski theorem (from 1924) states that {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} when {X} is a quadric, i.e., {X} is defined by just one polynomial equation of degree {2}.

Partly motivated by this, we introduce the following definition:

Definition 1 {X} satisfies Hasse’s principle (also called local-global principle) whenever {X(k)\neq\emptyset} if and only if {X(k_v)\neq\emptyset} for all places {v} of {k}.

As it turns out, Hasse’s principle is false in general: Swinnerton-Dyer constructed in 1962 some counterexamples among cubic surfaces, and Iskovskih constructed in 1970 a counterexample among the surfaces fibered in conics (given by intersections of two projective quadrics).

Of course, given that it is not hard to determine algorithmically when {X(k_v)\neq\emptyset} (with the help of Hensel lemma and/or Newton’s method), it is somewhat sad that Hasse’s principle fails in general.

In view of this state of affairs, we can try to generalize the problem of determining whether {X(k)\neq\emptyset} by replacing “rational points” by slightly more general objects (which then would be easier to find). In this direction, we have the following notion.

Definition 2 A zero-cycle {z} is a formal linear combination {z=\sum\limits_{x} n_x x} where:

  • {n_x\in\mathbb{Z}} vanishes for all but finitely many {x\in X}, and
  • if {n_x\neq 0}, then {x} is a closed point in the sense of Algebraic Geometry, i.e., {x} is a point defined over (its coordinates belong to) a finite extension {k(x)} of {k}.

The degree {deg(z)} of a zero-cycle {z} is {deg(z):=\sum\limits_{x} n_x [k(x):k]}.

Note that, by definition, a rational point {x\in X(k)} is a zero-cycle {z=x} of degree {deg(z) = 1}. Thus, we can ask the following more general question:

Does {X} possess a zero-cycle of degree {1} if {X} has such cycles over all {k_v}?

Remark 1 It follows from Bézout’s theorem that {X} has a zero-cycle of degree {1} if and only if {X} has points defined over finite extensions of {k} whose degrees are coprime.

Remark 2 A little curiosity about Bézout: as I discovered after moving from Paris to Avon, Bézout spent the last years of his life in Avon and the city gave his name to a street (not far from my appartment) in his honor.

Once more, the answer to this question is no: for example, it is known that there are counterexamples among surfaces fibered in conics.

Given this scenario, our goal is to explain how to refine the local-global principle with additional cohomological conditions (related to the so-called Brauer groups) introduced by Manin ensuring the existence of zero-cycles and/or rational points in certain situations.

Read More…

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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