arXiv subjects
Publications and source records attributed to Carlos Matheus.
Given an origami (square-tiled surface) M with automorphism group G, we compute the decomposition of the first homology group of M into isotypic G-submodules. Through the action of the affine group of M on the homology group, we deduce some consequences for the multiplicities of the Lyapunov exponents of the Kontsevich-Zorich cocycle. We also construct and study several families of interesting origamis illustrating our results.
We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale's Axiom A property is $C^1$-dense among the systems in this family, despite the existence of $C^2$-open subsets (closely related to the so-called Newhouse phenomena) where Smale's Axiom A is violated. In particular, this provides some evidence towards Smale's conjecture that Axiom A is a $C^1$-dense property among surface diffeomorphisms. The basic tools in the proof of this result are: 1) a recent theorem of Moreira saying that stable intersections of dynamical Cantor sets (one of the main obstructions to Axiom A property for surface diffeomorphisms) can be destroyed by $C^1$-perturbations; 2) the good geometry of the dynamical critical set (in the sense of Rodriguez-Hertz and Pujals) thanks to the particular form of Benedicks-Carleson toy models.
In the moduli space $H_g$ of normalized translation surfaces of genus $g$, consider, for a small parameter $ρ>0$, those translation surfaces which have two non-parallel saddle-connections of length $\leq ρ$. We prove that this subset of $H_g$ has measure $o(ρ^2)$ w.r.t. any probability measure on $H_g$ which is invariant under the natural action of $SL(2,R)$. This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunov exponents of the KZ-cocycle.
We prove that the critical Maxwell-Klein Gordon equation on R4+1 is globally well-posed for smooth initial data which are small in the energy. This reduces the problem of global regularity for large, smooth initial data to precluding concentration of energy.
We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (or Kodaira--Spencer map) of the Hodge bundle with respect to Gauss--Manin connection and investigate the relations between the central {Oseldets} subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.
We construct an explicit family of arithmetic Teichmüller curves $\mathcal{C}_{2k}$, $k\in\mathbb{N}$, supporting $\textrm{SL}(2,\mathbb{R})$-invariant probabilities $μ_{2k}$ such that the associated $\textrm{SL}(2,\mathbb{R})$-representation on $L^2(\mathcal{C}_{2k}, μ_{2k})$ has complementary series for every $k\geq 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves $\mathcal{C}_{2k}$ has arbitrarily slow rate of exponential mixing.
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 \cite{Forni:Matheus}) 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 found previously by Forni in genus three also corresponds to a square-tiled cyclic cover \cite{ForniSurvey}). We present several new examples of Teichmüller curves in strata of holomorphic and meromorphic quadratic differentials with maximally degenerate Kontsevich--Zorich spectrum. Presumably, these examples cover all possible Teichmüller curves with maximally degenerate spectrum. We prove that this is indeed the case within the class of square-tiled cyclic covers.
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-Matheus (in genus 4). We show that, in both cases, the action on the non trivial 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 non trivial Lyapunov exponents of the Kontsevich-Zorich cocycle for the Teichmuller disks of these two origamis are equal to zero.
We establish local and global well-posedness for the initial value problem associated to the one-dimensional Schrodinger-Debye (SD) system for data in the Sobolev spaces with low regularity. To obtain local results we prove two new sharp bilinear estimates for the coupling terms of this system in the continuous and periodic cases. Concerning global results, in the continuous case, the system is shown to be globally well-posed in $H^s\times H^s, -3/14< s< 0$. For initial data in Sobolev spaces with high regularity ($H^s\times H^s, s>5/2$), Bidégaray \cite{Bidegaray} proved that there are one-parameter families of solutions of the SD system converging to certain solutions of the cubic \emph{nonlinear Schrodinger equation} (NLS). Our results bellow $L^2\times L^2$ say that the SD system is not a good approach of the cubic NLS in Sobolev spaces with low regularity, since the cubic NLS is known to be ill-posed below $L^2$. The proof of our global result uses the \textbf{I}-method introduced by Colliander, Keel, Staffilani, Takaoka and Tao.
We construct an orientable holomorphic quadratic differential on a Riemann surface of genus 4 whose SL(2,R)-orbit is closed and has a highly degenerate Kontsevich - Zorich spectrum. This example is related to a previous similar construction in genus 3 by the first author.
Local and global well-posedness results are established for the initial value problem associated to the 1D Zakharov-Rubenchik system. We show that our results are sharp in some situations by proving Ill-posedness results otherwise. The global results allow us to study the norm growth of solutions corresponding to the Schrodinger equation term. We use ideas recently introduced to study the classical Zakharov systems.
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 \emph{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 {\it dnoidal}, and, moreover, we prove that all these periodic traveling waves are nonlinearly stable by perturbations with the same wavelength.
We study the existence, uniqueness and rate of decay of correlation of equilibrium measures associated to robust classes of non-uniformly expanding local diffeomorphisms and Hölder continuous potentials. The approach used in this paper is the spectral analysis of the Ruelle-Perron-Frobenius transfer operator. More precisely, we combine the expanding features of the eigenmeasures of the transfer operator with a Lasota-Yorke type inequality to prove the existence of an unique equilibrium measure with fast decay of correlations.
We prove that in a compact manifold of dimension $n\geq 2$, a $C^{1+α}$ volume-preserving diffeomorphisms that are robustly transitive in the $C^1$-topology have a dominated splitting. Also we prove that for 3-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field can not have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in 3-dimensional manifolds are Anosov. For this, we prove some ``pasting'' lemma, which allows to make perturbations in conservative systems.
We study local and global well-posedness of the initial value problem for the Schrödinger-Debye equation in the \emph{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 \emph{a priori} estimate for the $H^1$ norm of solutions of the periodic Schrödinger-Debye equation. A novel phenomena obtained as a by-product of this \emph{a priori} estimate is the global well-posedness of the periodic Schrödinger-Debye equation in dimensions $1,2$ and 3 \emph{without} any smallness hypothesis of the $H^1$ norm of the initial data in the ``focusing'' case.
Given a compact $n$-dimensional immersed Riemannian manifold $M^n$ in some Euclidean space we prove that if the Hausdorff dimension of the singular set of the Gauss map is small, then $M^n$ is homeomorphic to the sphere $S^n$. Also, we define a concept of finite geometrical type and prove that finite geometrical type hypersurfaces with small set of points of zero Gauss-Kronecker curvature are topologically the sphere minus a finite number of points. A characterization of the $2n$-catenoid is obtained.
In this note we construct measures of maximal entropy for a certain class of maps with critical points called Viana maps. The main ingredients of the proof are the non-uniform expansion features and the slow recurrence (to the critical set) of generic points with respect to the natural candidates for attaining the topological entropy.
We prove two new mixed sharp bilinear estimates of Schrödinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schrödinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our lowest regularity is $H^{1/4}\times L^2$, which is somewhat far from the naturally expected endpoint $L^2\times H^{-1/2}$. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint $L^2\times H^{-{3/4}+}$. Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space $H^1\times H^1$ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi.