arXiv subjects
Publications and source records attributed to Carlos Matheus.
This text is a slightly expanded version of a survey article on certain aspects of low dimensional dynamics and number theory written after a kind invitation by the editors of the Notices of the American Mathematical Society.
We analyse cusp excursions of random geodesics for Weil--Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions. We also give similar bounds for cusp excursions of random Weil--Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil--Petersson flow is polynomially mixing. Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.
We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ which is accumulated by a sequence of elements of the complement $M\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\setminus L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.
The (classical) Lagrange spectrum is a closed subset of the positive real numbers defined in terms of diophantine approximation. Its structure is quite involved. This article describes a polynomial time algorithm to approximate it in Hausdorff distance. It also extends to approximate the Markov spectrum related to infimum of binary quadratic forms.
We show that several portions of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ can be seen as subsets of arithmetic sums of Cantor sets with controlled Hausdorff dimensions. In particular, we prove that $M\setminus L$ has Hausdorff dimension strictly smaller than one.
The Lagrange and Markov spectra are classical objects in Number Theory related to certain Diophantine approximation problems. Geometrically, they are the spectra of heights of geodesics in the modular surface. These objects were first studied by A. Markov in 1879, but, despite many efforts, the structure of the complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ remained somewhat mysterious. In fact, it was shown by G. Freiman (in 1968 and 1973) and M. Flahive (in 1977) that $M\setminus L$ contains infinite \emph{countable} subsets near 3.11 and 3.29, and T. Cusick conjectured in 1975 that all elements of $M\setminus L$ were $<\sqrt{12}=3.46\dots$, and this was the \emph{status quo} of our knowledge of $M\setminus L$ until 2017. In this article, we show the following two results. First, we prove that $M\setminus L$ is \emph{richer} than it was previously thought because it contains a Cantor set of Hausdorff dimension larger than $1/2$ near $3.7$: in particular, this solves (negatively) Cusick's conjecture mentioned above. Secondly, we show that $M\setminus L$ is \emph{not} very thick: its Hausdorff dimension is strictly smaller than one.
We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $M\backslash L$ lying in distinct connected components of $\mathbb{R}\setminus L$, where $M$ is the Markov spectrum and $L$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k)_{k\in\mathbb{N}}$ of elements in $M$ converging to $3$ and we give some evidence towards the possibility that $m_k\in M\setminus L$ for all $k\geq 1$. In particular, this indicates that $3$ might belong to the closure of $M\setminus L$, so that the answer to Bousch's question about the closedness of $M\setminus L$ might be negative.
We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated to horseshoes with Hausdorff dimension $<22/21$ of conservative surface diffeomorphisms.
The Rauzy gasket $R$ is the maximal invariant set of a certain renormalization procedure for special systems of isometries naturally appearing in the context of Novikov's problem in conductivity theory for monocrystals. It was conjectured by Novikov and Maltsev in 2003 that the Hausdorff dimension $\dim_{\mathrm{H}}(R)$ of Rauzy gasket is strictly comprised between $1$ and $2$. In 2016, Avila, Hubert and Skripchenko confirmed that $\dim_{\mathrm{H}}(R)<2$. In this note, we use some results by Cao--Pesin--Zhao in order to show that $\dim_{\mathrm{H}}(R)>1.19$.
Let $G$ be a semisimple Lie group acting on a space $X$, let $μ$ be a compactly supported measure on $G$, and let $A$ be a strongly irreducible linear cocycle over the action of $G$. We then have a random walk on $X$, and let $T$ be the associated shift map. We show that the cocycle $A$ over the action of $T$ is conjugate to a block conformal cocycle. This statement is used in the recent paper by Eskin-Mirzakhani on the classifications of invariant measures for the SL(2,R) action on moduli space. The ingredients of the proof are essentially contained in the papers of Guivarch and Raugi and also Goldsheid and Margulis.
In this note, we exploit the arithmeticity criterion of Benoist--Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich--Zorich monodromy is not thin in the sense of Sarnak.
We extend some results of Marmi--Moussa--Yoccoz on the cohomological equations and local conjugacy classes of interval exchange maps of restricted Roth type. In particular, we answer a question of Krikorian about the codimension of the local conjugacy class of self-similar interval exchange maps associated to the Eierlegende Wollmilchsau and the Ornithorynque.
Filip showed that there are constants $C>0$ and $δ>0$ such that the number of special Lagrangian fibrations of volume $\leq V$ in a generic twistor family of K3 surfaces is $C\cdot V^{20}+O(V^{20-δ})$. In this note, we show that $δ$ can be taken to be any number $0<δ<\frac{4}{697633}$.
We describe the Kontsevich--Zorich cocycle over an affine invariant orbifold coming from a (cyclic) covering construction inspired by works of Veech and McMullen. In particular, using the terminology in a recent paper of Filip, we show that all cases of Kontsevich--Zorich monodromies of $SU(p,q)$ type are realized by appropriate covering constructions.
The complement $M\setminus L$ of the Lagrange spectrum $L$ in the Markov spectrum $M$ was studied by many authors (including Freiman, Berstein, Cusick and Flahive). After their works, we disposed of a countable collection of points in $M\setminus L$. In this article, we describe the structure of $M\setminus L$ near a non-isolated point $α_{\infty}$ found by Freiman in 1973, and we use this description to exhibit a concrete Cantor set $X$ whose Hausdorff dimension coincides with the Hausdorff dimension of $M\setminus L$ near $α_{\infty}$. A consequence of our results is the lower bound $HD(M\setminus L)>0.353$ on the Hausdorff dimension $HD(M\setminus L)$ of $M\setminus L$. Another by-product of our analysis is the explicit construction of new elements of $M\setminus L$, including its largest known member $c\in M\setminus L$ (surpassing the former largest known number $α_4\in M\setminus L$ obtained by Cusick and Flahive in 1989).
Freiman proved in 1968 that the Lagrange and Markov spectra do not coincide by exhibiting a countable infinite collection $\mathcal{F}$ of isolated points of the Markov spectrum which do not belong the Lagrange spectrum. In this paper, we describe the structure of the elements of the Markov spectrum in the largest interval $(c_{\infty}, C_{\infty})$ containing $\mathcal{F}$ and avoiding the Lagrange spectrum. In particular, we compute the smallest known element $f$ of $M\setminus L$, and we show that the Hausdorff dimension of the portion of the Markov spectrum between $c_{\infty}$ and $C_{\infty}$ is $> 0.2628$.
Let $φ_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $Λ_0$ be a horseshoe of $φ_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a small smooth area-preserving perturtabion $φ$ of $φ_0$, let $L_{φ, f}$, resp. $M_{φ, f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the $φ$-orbits of points in the horseshoe $Λ$ obtained by hyperbolic continuation of $Λ_0$. We show that, for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ vary continuously with $t\in\mathbb{R}$ and, moreover, $M_{φ, f}\cap (-\infty, t)$ has the same Hausdorff dimension of $L_{φ, f}\cap (-\infty, t)$ for all $t\in\mathbb{R}$.
We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus for a set of (large) parameters of positive Lebesgue measure.