Concentration of dimension in the Lagrange spectrum

TL;DR

This paper investigates the structure and dimension of the dynamical Lagrange spectrum associated with certain surface diffeomorphisms, revealing how the spectrum's local Hausdorff dimension behaves under specific conditions.

Contribution

It introduces a generalized Diophantine approximation constant for points in a dynamical setting and analyzes the Hausdorff dimension of the spectrum, extending classical results to a dynamical context.

Findings

Identifies conditions where the Hausdorff dimension of level sets of the spectrum coincide.

Describes the local Hausdorff dimension of the dynamical Lagrange spectrum when the Hausdorff dimension of the horseshoe is less than one.

Recovers classical Lagrange spectrum results within the dynamical framework.

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.