Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra

Abstract

Let $\varphi_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $\Lambda_0$ 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 $\Lambda$ obtained by hyperbolic continuation of $\Lambda_0$. We show that, for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-\infty, t)$ vary continuously with $t\in\mathbb{R}$ and, moreover, $M_{\varphi, f}\cap (-\infty, t)$ has the same Hausdorff dimension of $L_{\varphi, f}\cap (-\infty, t)$ for all $t\in\mathbb{R}$.