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

Abstract

Let ${\phi }_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let ${\Lambda }_0$ be a transitive horseshoe of ${\phi }_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $\phi$ of ${\phi }_0$, let $L_{\phi ,f}$ and $M_{\phi ,f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $\Lambda (\phi )$ of the horseshoe ${\Lambda }_0$ and $f$. We show that for generic choices of $\phi$ and $f$, the Hausdorff dimension of the sets $L_{\phi ,f}\cap (-\infty ,t)$ and $M_{\phi ,f}\cap (-\infty ,t)$ 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.

24 pages,2 figures