Concentration of dimension in the Lagrange spectrum

Abstract

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