Concentration of dimension in the Lagrange spectrum

Abstract

Let $\varphi$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda$ be a mixing horseshoe of $\varphi$. Given a smooth real function $f$ defined on $S$, we define for points $\eta$ in the unstable Cantor set of the pair $(\varphi,\Lambda)$, a generalization, $k_{\varphi,\Lambda,f}(\eta)$, of the best constant of Diophantine approximation for irrational numbers. We study the set of points $\eta$ for which the sets $k_{\varphi,\Lambda,f}^{-1}((-\infty,\eta])$ and $k_{\varphi,\Lambda,f}^{-1}(\eta)$ have the same Hausdorff dimension and when the Hausdorff dimension of $\Lambda$ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, $\mathcal{L}_{\varphi,\Lambda,f}$, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra.