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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.
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  • 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.


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

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