On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Abstract

Let ${\phi }_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let ${\Lambda }_0$ be a mixing horseshoe of ${\phi }_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $\phi$, close to ${\phi }_0$, let ${\mathcal{L}}_{\phi ,f}$ be the Lagrange spectrum associated to the hyperbolic continuation $\Lambda (\phi )$ of the horseshoe ${\Lambda }_0$ and $f$. We show that, for generic choices of $\phi$ and $f$, if $L_{\phi ,f}$ is the map that gives the Hausdorff dimension of the set ${\mathcal{L}}_{\phi ,f}\cap (-\infty ,t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{\phi ,f}$.