Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra
Let $φ_0$ be a smooth area-preserving diffeomorphism of a compact surface $M$ and let $Λ_0$ be a horseshoe of $φ_0$ with Hausdorff dimension strictly smaller than one. Given a smooth function $f:M\to \mathbb{R}$ and a small smooth area-preserving perturtabion $φ$ of $φ_0$, let $L_{φ, f}$, resp. $M_{φ, f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the $φ$-orbits of points in the horseshoe $Λ$ obtained by hyperbolic continuation of $Λ_0$. We show that, for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ vary continuously with $t\in\mathbb{R}$ and, moreover, $M_{φ, f}\cap (-\infty, t)$ has the same Hausdorff dimension of $L_{φ, f}\cap (-\infty, t)$ for all $t\in\mathbb{R}$.