Continuity of Hausdorff Dimension Across Generic Dynamical Lagrange and Markov Spectra II

TL;DR

This paper proves that for generic perturbations of negatively curved surfaces, the Hausdorff dimension of certain dynamical spectra varies continuously and matches between Lagrange and Markov spectra, extending previous results.

Contribution

It establishes the continuity and equality of Hausdorff dimensions of Lagrange and Markov spectra for generic perturbations of negatively curved surfaces.

Findings

Hausdorff dimension of spectra varies continuously with t

Hausdorff dimensions of Lagrange and Markov spectra coincide

Results hold for generic smooth perturbations

Abstract

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface $N$, and let $\Lambda_0$ be a basic hyperbolic set of the geodesic flow of $g_0$ with Hausdorff dimension strictly smaller than two. Given a small smooth perturbation $g$ of $g_0$ and a smooth real-valued function $f$ on the unit tangent bundle to $N$ with respect to $g$, let $L_{g,\Lambda,f}$, resp. $M_{g,\Lambda,f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the geodesics in the hyperbolic continuation $\Lambda$ of $\Lambda_0$. We prove that, for generic choices of $g$ and $f$, the Hausdorff dimension of the sets $L_{g,\Lambda, f}\cap (-\infty, t)$ vary continuously with $t\in\mathbb{R}$ and, moreover, $M_{g,\Lambda, f}\cap (-\infty, t)$ has the same Hausdorff dimension of $L_{g,\Lambda, f}\cap (-\infty, t)$ for all $t\in\mathbb{R}$.