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

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

Aline Cerqueira 1, , Carlos Matheus 2, and  Carlos Gustavo Moreira 1,

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

Fund Project: 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.

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.
Mathematics Subject Classification: Primary: 11J06, 37E30; Secondary: 37D05.
Citation: Aline Cerqueira, Carlos Matheus, Carlos Gustavo Moreira. Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra. Journal of Modern Dynamics, 2018, 12: 151-174. doi: 10.3934/jmd.2018006
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.  Google Scholar

[2]

S. Hersonsky and F. Paulin, Diophantine approximation for negatively curved manifolds, Math. Z., 241 (2002), 181-226.  doi: 10.1007/s002090200412.  Google Scholar

[3]

M. Hirsch, C Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

S. Hersonsky and F. Paulin, Diophantine approximation for negatively curved manifolds, Math. Z., 241 (2002), 181-226.  doi: 10.1007/s002090200412.  Google Scholar

[3]

M. Hirsch, C Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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