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Carlos Matheus

Publications and source records attributed to Carlos Matheus.

At least 55 records · Page 3Linked to original sources
arXiv2017

Zorich conjecture for hyperelliptic Rauzy-Veech groups

We describe the structure of hyperelliptic Rauzy diagrams and hyperelliptic Rauzy-Veech groups. In particular, this provides a solution of the hyperelliptic cases of a conjecture of Zorich on the Zariski closure of Rauzy-Veech groups.

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arXiv2017

The Lagrange and Markov spectra from the dynamical point of view

This text grew out of my lecture notes for a 4-hours minicourse delivered on October 17 \& 19, 2016 during the research school "Applications of Ergodic Theory in Number Theory" -- an activity related to the Jean-Molet Chair project of Mariusz Lemańczyk and Sébastien Ferenczi -- realized at CIRM, Marseille, France. The subject of this text is the same of my minicourse, namely, the structure of the so-called Lagrange and Markov spectra (with an special emphasis on a recent theorem of C. G. Moreira).

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arXiv2016

Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities

This work constructs symbolic dynamics for non-uniformly hyperbolic surface maps with a set of discontinuities $D$. We allow the derivative of points nearby $D$ to be unbounded, of the order of a negative power of the distance to $D$. Under natural geometrical assumptions on the underlying space $M$, we code a set of non-uniformly hyperbolic orbits that do not converge exponentially fast to $D$. The results apply to non-uniformly hyperbolic planar billiards, e.g. Bunimovich stadia.

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arXiv2016

Rates of mixing for the Weil-Petersson geodesic flow II: exponential mixing in exceptional moduli spaces

We establish exponential mixing for the geodesic flow $φ_t\colon T^1S\to T^1S$ of an incomplete, negatively curved surface $S$ with cusp-like singularities of a prescribed order. As a consequence, we obtain that the Weil-Petersson flows for the moduli spaces ${\mathcal M}_{1,1}$ and ${\mathcal M}_{0,4}$ are exponentially mixing, in sharp contrast to the flows for ${\mathcal M}_{g,n}$ with $3g-3+n>1$, which fail to be rapidly mixing. In the proof, we present a new method of analyzing invariant foliations for hyperbolic flows with singularities, based on changing the Riemannian metric on the phase space $T^1S$ and rescaling the flow $φ_t$.

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arXiv2016

Lecture notes on the dynamics of the Weil-Petersson flow

This text grew out of some lecture notes prepared by the author in the occasion of a series of three lectures during the workshop "Young mathematicians in dynamical systems" organized by Francoise Dal'bo, Louis Funar, Boris Hasselblatt and Barbara Schapira in November 2013 at CIRM, Marseille, France. The three lectures at the origin of this text were part of a minicourse by Keith Burns, Boris Hasselblatt and the author around the recent theorem of Burns-Masur-Wilkinson on the ergodicity of the Weil-Petersson (WP) geodesic flow. The goal of these notes is the same of the author's lectures: we cover some of the aspects related to moduli spaces of Riemann surfaces (and Teichmueller theory) in the Burns-Masur-Wilkinson's proof of the ergodicity of WP flow.

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arXiv2015

Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups

We give an example of a Teichmüller curve which contains, in a factor of its monodromy, a group which was not observed before. Namely, it has Zariski closure equal to the group $SO^*(6)$ in its standard representation; up to finite index, this is the same as $SU(3,1)$ in its second exterior power representation. The example is constructed using origamis (i.e. square-tiled surfaces). It can be generalized to give monodromy inside the group $SO^*(2n)$ for all $n$, but in the general case the monodromy might split further inside the group. Also, we take the opportunity to compute the multiplicities of representations in the (0,1) part of the cohomology of regular origamis, answering a question of Matheus-Yoccoz-Zmiaikou.

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arXiv2014

A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $SL_2(R)$-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M. Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Lelièvre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus $>0$) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

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arXiv2014

A coding-free simplicity criterion for the Lyapunov exponents of Teichmueller curves

In this note we show that the results of H. Furstenberg on the Poisson boundary of lattices of semisimple Lie groups allow to deduce simplicity properties of the Lyapunov spectrum of the Kontsevich-Zorich cocycle of Teichmueller curves in moduli spaces of Abelian differentials without the usage of codings of the Teichmueller flow. As an application, we show the simplicity of some Lyapunov exponents in the setting of (some) Prym Teichmueller curves of genus 4 where a coding-based approach seems hard to implement because of the poor knowledge of the Veech group of these Teichmueller curves. Finally, we extend the discussion in this note to show the simplicity of Lyapunov exponents coming from (high weight) variations of Hodge structures associated to mirror quintic Calabi-Yau threefolds.

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arXiv2014

Zero Lyapunov exponents of the Hodge bundle

By the results of G. Forni and of R. Treviño, the Lyapunov spectrum of the Hodge bundle over the Teichmüller geodesic flow on the strata of Abelian and of quadratic differentials does not contain zeroes even though for certain invariant submanifolds zero exponents are present in the Lyapunov spectrum. In all previously known examples, the zero exponents correspond to those PSL(2,R)-invariant subbundles of the real Hodge bundle for which the monodromy of the Gauss-Manin connection acts by isometries of the Hodge metric. We present an example of an arithmetic Teichmüller curve, for which the real Hodge bundle does not contain any PSL(2,R)-invariant, continuous subbundles, and nevertheless its spectrum of Lyapunov exponents contains zeroes. We describe the mechanism of this phenomenon; it covers the previously known situation as a particular case. Conjecturally, this is the only way zero exponents can appear in the Lyapunov spectrum of the Hodge bundle for any PSL(2,R)-invariant probability measure.

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arXiv2014

Un contre-exemple à la réciproque du critère de Forni pour la positivité des exposants de Lyapunov du cocycle de Kontsevich-Zorich

We introduce two square-tiled surfaces, one with $8$ squares inside $Ω\mathcal{M}_3(2,2)$, and the other with $9$ squares inside $Ω\mathcal{M}_4(3,3)$, respectively. In these examples, the dimensions of the isotropic subspaces (in absolute homology) generated by the waist curves of the maximal cylinders in any fixed rational direction are $2$ and $3$ respectively. Hence, a geometrical criterion of G. Forni for the non-uniform hyperbolicity of Kontsevich-Zorich (KZ) cocycle can not be applied to these examples. Nevertheless, we prove that there are no vanishing exponents and the spectrum is simple for these two square-tiled surfaces. In particular, the non-vanishing of exponents of KZ cocycle for a regular measure doesn't imply that the support of this measure contains a completely periodic surface whose waist curves of maximal cylinders generates a Lagrangian subspace in its absolute homology.

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arXiv2013

Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Bedlewo school "Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory" (from 4 to 16 July, 2011). In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the SL(2,R)-action on these moduli spaces and the Kontsevich-Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich-Zorich cocycle, with respect to Masur-Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich-Zorich cocycle work as \emph{renormalization dynamics} for interval exchange transformations and translation flows. In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich-Zorich cocycle with respect to invariant measures other than the Masur-Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich-Zorich cocycle are very different from the case of Masur-Veech measures. Finally, we end these notes by constructing some examples of closed SL(2,R)-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary SL(2,R)-representations have arbitrarily small spectral gap (and in particular it has complementary series).

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arXiv2013

Fractal geometry of non-uniformly hyperbolic horseshoes

This article is an expanded version of some notes for my talk at the ``Ergodic Theory and Dynamical Systems Workshop'' (from March 22 to March 25, 2012) held at the Department of Mathematics of the University of North Carolina at Chapel Hill. In the aforementioned talk, it was discussed some recent results on the fractal geometry of certain objects -- \emph{non-uniformly hyperbolic horseshoes} -- constructed by Jacob Palis and Jean-Christophe Yoccoz in their recent \emph{tour-de-force} work around \emph{heteroclinic bifurcations of surface diffeomorphisms}. The goal of the present article is two-fold. The first part will be a modest survey on the history of the study of homoclinic/heteroclinic bifurcations of surface diffeomorphisms: starting from the seminal works of Henri Poincaré on Celestial Mechanics, we will recall some landmark results on bifurcations of homoclinic/heteroclinic tangencies associated to \emph{uniformly hyperbolic horseshoes}. Our discussions in this (first) part will culminate with a brief presentation of the scheme of the proof of the main theorems of J. Palis and J.-C. Yoccoz on this subject. In particular, we will highlight the main features of the so-called non-uniformly hyperbolic horseshoes, an object at the heart of the work of J. Palis and J.-C. Yoccoz. Then, the second part will be a sort of research announcement where we will discuss some results obtained in collaboration with J. Palis and J.-C. Yoccoz on the (fractal) geometry of non-uniformly hyperbolic horseshoes.

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