On the minima of Markov and Lagrange Dynamical Spectra

TL;DR

This paper proves that for typical functions on surfaces with horseshoe dynamics, the minima of Lagrange and Markov spectra coincide and are achieved at periodic points, answering a question by Yoccoz.

Contribution

It establishes the equality of minima of Lagrange and Markov spectra for a broad class of functions on surface horseshoes, solving a longstanding open problem.

Findings

Minima of spectra coincide for typical functions.

Minima are attained at periodic points.

Results apply to a large set of functions on surfaces.

Abstract

We consider typical Lagrange and Markov dynamical spectra associated to horseshoes on surfaces. We show that for a large set of real functions on the surface, the minima of the corresponding Lagrange and Markov dynamical spectra coincide and are given by the image of a periodic point of the dynamics by the real function. This solves a question by Jean-Christophe Yoccoz.