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

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

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 $t∈\mathbb{R}$.

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.

[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.

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.

[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.

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