Article Contents
Article Contents

Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

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 $t∈\mathbb{R}$.

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

 Citation:

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

•  [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.

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