Continuity of Hausdorff dimension across generic dynamical Lagrange and Markov spectra II

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 $L_{g,\Lambda ,f}$ (respectively $M_{g,\Lambda ,f}$ ) be the Lagrange (respectively Markov) spectrum of asymptotic highest (respectively 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 dimensions 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 as $L_{g,\Lambda , f}\cap (-\infty , t)$ for all $t\in \mathbb {R}$ .

On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

“…The continuity of the Hausdorff dimension of the intersection of the spectra with the half-line (−∞, t ) was studied in [8] for geodesic flows on surfaces of negative curvature, generalizing results of [25,7].…”

“…The equation (8) implies that the functions ξ 1 • π and ξ 2 • π have disjoint support and therefore the perturbation of f , f 1 , does not affect the perturbation f 2 . Thus, as it is explained in [34,Lemma 4.7], we can perform a small perturbation, f , of f on V in such a way that 0 is a simultaneous regular value of the functions…”

On the Lagrange and Markov Dynamical Spectra for Anosov Flows in Dimension 3

“…Many recent results, such as [8,19], and [9], show that the Lagrange and Markov spectra are well behaved when looking at the hyperbolic world, including geodesic flows of negative curvature (cf. [23]), Teichmüller flows (cf.…”