Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra

Abstract

Let $\varphi_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a transitive horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $\varphi$ of $\varphi_0$, let $L_{\varphi, f}$ and $M_{\varphi, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $\Lambda(\varphi)$ of the horseshoe $\Lambda_0$ and $f$. We show that for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-\infty, t)$ and $M_{\varphi, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension.