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

Let $g_0$ be a smooth pinched negatively curved Riemannian metric on a complete surface $N$, and let ${\mathrm{\Lambda }}_0$ 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 $L_{g,\mathrm{\Lambda },f}$, resp. $M_{g,\mathrm{\Lambda },f}$ be the Lagrange, resp. Markov spectrum of asymptotic highest, resp. highest values of $f$ along the geodesics in the hyperbolic continuation $\mathrm{\Lambda }$ of ${\mathrm{\Lambda }}_0$. We prove that, for generic choices of $g$ and $f$, the Hausdorff dimension of the sets $L_{g,\mathrm{\Lambda },f}\cap (-\mathrm{\infty },t)$ vary continuously with $t\in \mathbb{R}$ and, moreover, $M_{g,\mathrm{\Lambda },f}\cap (-\mathrm{\infty },t)$ has the same Hausdorff dimension of $L_{g,\mathrm{\Lambda },f}\cap (-\mathrm{\infty },t)$ for all $t\in \mathbb{R}$.