arXiv subjects
Publications and source records attributed to Carlos Matheus.
Almost 20 years ago, the first and fourth authors found examples of SL(2,R)-invariant subbundles of Hodge bundles over Teichm\"uller curves having maximally degenerate Lyapunov spectrum. For these same surfaces, we show that a natural non-Abelian analogue has no zero Lyapunov exponents.
The $SL(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis can be represented by finite four-regular graphs. It is a conjecture of McMullen that the orbit graphs of such origamis in the stratum $\mathcal{H}(2)$ form an expander family. We provide indirect evidence for this conjecture by proving that the absolute values of the Euler characteristics of the graphs in this family go to infinity with the number of squares $n$. This generalises previous work of the authors, in which we established eventual non-planarity for this family, and provides the strongest indirect evidence to date for McMullen's conjecture. We also prove that the same phenomenon holds for primitive origamis in the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$. Assuming conjectures of Zmiaikou and Delecroix--Leli\`evre, we establish the same in $\mathcal{H}(1,1)$ and for two families of non-Prym origamis in $\mathcal{H}(4)$. Finally, assuming a stronger conjecture concerning orbit growth in low-complexity strata, we establish that for any family of $SL(2,\mathbb{Z})$-orbit graphs of primitive origamis in a stratum with one or two conical singularities, the absolute values of the Euler characteristics of the graphs go to infinity with the number of squares along a density one subsequence. By relating the genus of the elliptic generator orbit graphs to the genus of the associated arithmetic Teichm\"uller curves, we also recover results of Mukamel and extend results of Torres-Teigel--Zachhuber establishing that the genus of these Teichm\"uller curves in $\mathcal{H}(2)$ and the Prym loci of $\mathcal{H}(4)$ and $\mathcal{H}(6)$ also go to infinity. The proofs rely on counts of integral points on algebraic hypersurfaces using methods of Bombieri--Pila and Browning--Gorodnik, on counts of orbifold points on Teichm\"uller curves, and on counts of pseudo-Anosov diffeomorphisms with bounded dilatation.
We prove a nonstandard central limit theorem and weak invariance principle, with superdiffusive normalisation $(t\log t)^{1/2}$, for geodesic flows on a class of nonpositively curved surfaces with flat cylinder. We also prove that correlations decay at rate $t^{-1}$. An important ingredient of the proof, which is of independent interest, is an improved results on the regularity of the stable/unstable foliations induced by the Green bundles.
We consider a class of nonuniformly hyperbolic dynamical systems with a first return time satisfying a central limit theorem (CLT) with nonstandard normalisation $(n\log n)^{1/2}$. For such systems (both maps and flows) we show that it automatically follows that the functional central limit theorem or weak invariance principle (WIP) with normalisation $(n\log n)^{1/2}$ holds for H\"older observables. Our approach streamlines certain arguments in the literature. Applications include various examples from billiards, geodesic flows and intermittent dynamical systems. In this way, we unify existing results as well as obtaining new results. In particular, we deduce the WIP with nonstandard normalisation for Bunimovich stadia as an immediate consequence of the corresponding CLT proved by B\'alint & Gou\"ezel.
Using algorithms implicit in the classification of $SL(2,\mathbb{Z})$-orbits of primitive origamis in the stratum $\mathcal{H}(2)$ due to Hubert-Leli\`evre and McMullen, we give diameter bounds on the resulting orbit graphs. Since the machinery of McMullen from $\mathcal{H}(2)$ is generalised and reused in Lanneau and Nguyen's classification of the orbits of Prym eigenforms in $\mathcal{H}(4)$ and $\mathcal{H}(6),$ we are also able to obtain diameter bounds for the orbit graphs in this setting as well. In each stratum, we obtain diameter bounds of the form $O(N^{2/3}\log N)$, where $N$ is the size of the orbit graph.
We use the Selberg zeta function to study the limit behavior of resonances in a degenerating family of Kleinian Schottky groups. We prove that, after a suitable rescaling, the Selberg zeta functions converge to the Ihara zeta function of a limiting finite graph associated to the relevant non-Archimedean Schottky group acting on the Berkovich projective line. Moreover, we show that these techniques can be used to get an exponential error term in a result of McMullen (recently extended by Dang and Mehmeti) about the asymptotics for the vanishing rate of the Hausdorff dimension of limit sets of certain degenerating Schottky groups generating symmetric three-funnel surfaces. Here, one key idea is to introduce an intermediate zeta function capturing \emph{both} non-Archimedean and Archimedean information (while the traditional Selberg, resp. Ihara zeta functions concern only Archimedean, resp. non-Archimedean properties).
The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$. In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--W\"ustholz theorem on linear forms in logarithms of algebraic numbers.
We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for $area^3$ Siegel-Veech constants is always equal to the reciprocal of the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.
Let $\Omega$ be a strictly convex divisible subset of the $n$-dimensional real projective space which is not an ellipsoid. Even though $\partial\Omega$ is not $C^2$, Benoist showed that it is $C^{1+\alpha}$ for some $\alpha>0$, and Crampon established that $\partial\Omega$ actually possesses a sort of anisotropic H\"older regularity -- described by a list $\alpha_1\leq\dots\leq\alpha_{n-1}$ of positive real numbers -- at almost all of its points. In this article, we show that $\partial\Omega$ is maximally anisotropic in the sense that this list of approximate regularities of $\partial\Omega$ does not contain repetitions. This result is a consequence of the simplicity of the Lyapunov spectrum of the Hilbert geodesic flow for every equilibrium measure associated to a H\"older potential.
The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition $(\star)$ is fulfilled. In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus $7$, resp. $10$.
In this note, we extend the scope of our previous work joint with Bonnafoux, Kattler, Ni\~no, Sedano-Mendoza, Valdez and Weitze-Schmith\"usen by showing the arithmeticity of the Kontsevich--Zorich monodromies of infinite families of square-tiled surfaces of genera four, five and six.
We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.
Let $L$ and $M$ denote the Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. In this work, we exhibit new gaps of $L$ and $M$ using two methods. First, we derive such gaps by describing a new portion of $M\setminus L$ near to 3.938: this region (together with three other candidates) was found by investigating the pictures of $L$ recently produced by V. Delecroix and the last two authors with the aid of an algorithm explained in one of the appendices to this paper. As a by-product, we also get the largest known elements of $M\setminus L$ and we improve upon a lower bound on the Hausdorff dimension of $M\setminus L$ obtained by the last two authors together with M. Pollicott and P. Vytnova (heuristically, we get a new lower bound of $0.593$ on the dimension of $M\setminus L$). Secondly, we use a renormalisation idea and a thickness criterion (reminiscent from the third author's PhD thesis) to detect infinitely many maximal gaps of $M$ accumulating to Freiman's gap preceding the so-called Hall's ray $[4.52782956616...,\infty)\subset L$.
Utilizing KAM theory, we show that there are certain levels in relative SU(2) and SU(3) character varieties of the once-punctured torus where the action of a single hyperbolic element is not ergodic.
The variations of Hodge structures of weight one associated to square-tiled surfaces naturally generate interesting subgroups of integral symplectic matrices called Kontsevich--Zorich monodromies. In this paper, we show that arithmetic groups are frequent among the Kontsevich--Zorich monodromies of square-tiled surfaces of low genera $g$.
We prove polynomial decay of correlations for geodesic flows on a class of nonpositively curved surfaces where zero curvature only occurs along one closed geodesic. We also prove that various statistical limit laws, including the central limit theorem, are satisfied by this class of geodesic flows.
We consider the $\text{SL}(2,\mathbb{Z})$-orbits of primitive $n$-squared origamis in the stratum $\mathcal{H}(2)$. In particular, we consider the 4-valent graphs obtained from the action of $\text{SL}(2,\mathbb{Z})$ with respect to a generating set of size two. We prove that, apart from the orbit for $n = 3$ and one of the orbits for $n = 5$, all of the obtained graphs are non-planar. Specifically, in each of the graphs we exhibit a $K_{3,3}$ minor, where $K_{3,3}$ is the complete bipartite graph on two sets of three vertices.
This paper is dedicated to the study of two famous subsets of the real line, namely Lagrange spectrum $L$ and Markov spectrum $M$. Our first result, Theorem 2.1, provides a rigorous estimate on the smallest value $t_1$ such that the portion of the Markov spectrum $(-\infty,t_1)\cap M$ has Hausdorff dimension $1$. Our second result, Theorem 3.1, gives a new upper bound on the Hausdorff dimension of the set difference $M\setminus L$. Our method combines new facts about the structure of the classical spectra together with finer estimates on the Hausdorff dimension of Gauss--Cantor sets of continued fraction expansions whose entries satisfy appropriate restrictions.