Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

Aline Cerqueira; Carlos Matheus; Carlos Gustavo Moreira

doi:10.3934/jmd.2018006

Article Contents

2018, Volume 12: 151-174. Doi: 10.3934/jmd.2018006

This volume Previous Article Distortion and the automorphism group of a shift Next Article Symmetry of entropy in higher rank diagonalizable actions and measure classification

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

1.
IMPA, Instituto de Matemática Pura e Aplicada, Estrada Dona Castorina, Jardim Botânico, Rio de Janeiro, RJ, CEP 22460-320, Brazil
2.
Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93439, Villetaneuse, France

Received: February 25, 2017

Revised: February 18, 2018

Published: April 2018

AC: Partially supported by CNPq-Brazil. Also, she thanks the hospitality of Collège de France and IMPA-Brazil during the preparation of this article.
CM: Temporarily affiliated to the UMI CNRS-IMPA (UMI 2924) during the final stages of preparation of this work and he is grateful to IMPA-Brazil for the hospitality during this period.
CGM: Partially supported by CNPq-Brazil.

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 11J06, 37E30; Secondary: 37D05.

Citation:

Full Text(HTML)

Figure 1. Geometry of the horseshoe $\Lambda$.

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	T. Cusick and M. Flahive, The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, Providence, RI, 1989. doi: 10.1090/surv/030.
[2]	S. Hersonsky and F. Paulin, Diophantine approximation for negatively curved manifolds, Math. Z., 241 (2002), 181-226. doi: 10.1007/s002090200412.
[3]	M. Hirsch, C Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.
[4]	C. G. Moreira, Geometric properties of the Markov and Lagrange spectra, preprint, 2016, available at arXiv: 1612.05782, accepted for publication in Ann. Math.
[5]	C. G. Moreira, Geometric properties of images of cartesian products of regular Cantor sets by differentiable real maps, preprint, 2016, available at arXiv: 1611.00933.
[6]	C. G. Moreira and S. Romaña, On the Lagrange and Markov dynamical spectra, Ergodic Theory Dynam. Systems, 37 (2017), 1570-1591. doi: 10.1017/etds.2015.121.
[7]	C. G. Moreira and J.-C. Yoccoz, Tangences homoclines stables pour des ensembles hyperboliques de grande dimension fractale, Ann. Sci. Éc. Norm. Supér. (4), 43 (2010), 1-68. doi: 10.24033/asens.2115.
[8]	J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, 35 Cambridge University Press, Cambridge, 1993.