CPAA
On the periodic Schrödinger-Debye equation
Alexander Arbieto Carlos Matheus
Communications on Pure & Applied Analysis 2008, 7(3): 699-713 doi: 10.3934/cpaa.2008.7.699
We study local and global well-posedness of the initial value problem for the Schrödinger-Debye equation in the periodic case. More precisely, we prove local well-posedness for the periodic Schrödinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new a priori estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomenon obtained as a by-product of this a priori estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1$ and $2$ without any smallness hypothesis of the $H^1$ norm of the initial data in the "focusing" case.
keywords: Schrödinger-Debye system Bourgain's method. well-posedness
JMD
Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards
Giovanni Forni Carlos Matheus
Journal of Modern Dynamics 2014, 8(3&4): 271-436 doi: 10.3934/jmd.2014.8.271
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 Będlewo 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 $SL(2,\mathbb{R})$-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 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 $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).
keywords: \mathbb{R})$-action on moduli spaces Moduli spaces Kontsevich–Zorich cocycle Abelian differentials translation surfaces Teichmüller flow $SL(2 Lyapunov exponents.
DCDS
An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes
Carlos Matheus Jacob Palis
Discrete & Continuous Dynamical Systems - A 2018, 38(2): 431-448 doi: 10.3934/dcds.2018020

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.

keywords: Heteroclinic bifurcations non-uniformly hyperbolic horseshoes Hausdorff dimension
JMD
The action of the affine diffeomorphisms on the relative homology group of certain exceptionally symmetric origamis
Carlos Matheus Jean-Christophe Yoccoz
Journal of Modern Dynamics 2010, 4(3): 453-486 doi: 10.3934/jmd.2010.4.453
We compute explicitly the action of the group of affine diffeomorphisms on the relative homology of two remarkable origamis discovered respectively by Forni (in genus $3$) and Forni and Matheus (in genus $4$). We show that, in both cases, the action on the nontrivial part of the homology is through finite groups. In particular, the action on some $4$-dimensional invariant subspace of the homology leaves invariant a root system of $D_4$ type. This provides as a by-product a new proof of (slightly stronger versions of) the results of Forni and Matheus: the nontrivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmüller disks of these two origamis are equal to zero.
keywords: Kontsevich-Zorich cocycle totally degenerate origamis. Teichmüller dynamics
JMD
Square-tiled cyclic covers
Giovanni Forni Carlos Matheus Anton Zorich
Journal of Modern Dynamics 2011, 5(2): 285-318 doi: 10.3934/jmd.2011.5.285
A cyclic cover of the complex projective line branched at four appropriate points has a natural structure of a square-tiled surface. We describe the combinatorics of such a square-tiled surface, the geometry of the corresponding Teichmüller curve, and compute the Lyapunov exponents of the determinant bundle over the Teichmüller curve with respect to the geodesic flow. This paper includes a new example (announced by G. Forni and C. Matheus in [17] of a Teichmüller curve of a square-tiled cyclic cover in a stratum of Abelian differentials in genus four with a maximally degenerate Kontsevich--Zorich spectrum (the only known example in genus three found previously by Forni also corresponds to a square-tiled cyclic cover [15]. We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with a maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectra. We prove that this is indeed the case within the class of square-tiled cyclic covers.
keywords: Kontsevich--Zorich cocycle Teichmüller geodesic flow square-tiled surfaces.
CPAA
Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system
Jaime Angulo Carlos Matheus Didier Pilod
Communications on Pure & Applied Analysis 2009, 8(3): 815-844 doi: 10.3934/cpaa.2009.8.815
The objective of this paper is two-fold: firstly, we develop a local and global (in time) well-posedness theory for a system describing the motion of two fluids with different densities under capillary-gravity waves in a deep water flow (namely, a Schrödinger-Benjamin-Ono system) for low-regularity initial data in both periodic and continuous cases; secondly, a family of new periodic traveling waves for the Schrödinger-Benjamin-Ono system is given: by fixing a minimal period we obtain, via the implicit function theorem, a smooth branch of periodic solutions bifurcating a Jacobian elliptic function called dnoidal, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.
keywords: Nonlinear PDE traveling wave solutions. initial value problem
JMD
Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra
Aline Cerqueira Carlos Matheus Carlos Gustavo Moreira
Journal of Modern Dynamics 2018, 12(1): 151-174 doi: 10.3934/jmd.2018006

Let $\varphi_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $Λ_0$ be a horseshoe of $\varphi_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a small smooth area-preserving perturtabion $\varphi$ of $\varphi_0$, let $L_{\varphi, f}$, resp. $M_{\varphi, f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the $\varphi$-orbits of points in the horseshoe $Λ$ obtained by hyperbolic continuation of $Λ_0$.

We show that, for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-∞, t)$ vary continuously with $t∈\mathbb{R}$ and, moreover, $M_{\varphi, f}\cap (-∞, t)$ has the same Hausdorff dimension as $L_{\varphi, f}\cap (-∞, t)$ for all $t∈\mathbb{R}$.

keywords: Hausdorff dimension horseshoes Lagrange spectrum Markov spectrum surface diffeomorphisms

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