Concentration of dimension in the Lagrange spectrum

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