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

Abstract

Let $φ_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $φ$ of $φ_0$, let $L_{φ, f}$ and $M_{φ, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ and $M_{φ, f}\cap (-\infty, t)$ 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.