On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Abstract

Let $φ_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a mixing horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $φ$, close to $φ_0$, let $\mathcal{L}_{φ, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that, for generic choices of $φ$ and $f$, if $L_{φ, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{φ, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{φ, f}$.