On the Lagrange and Markov Dynamical Spectra for Geodesic Flows in Surfaces with Negative Curvature

TL;DR

This paper demonstrates that for typical negatively curved surfaces, the Lagrange and Markov dynamical spectra associated with geodesic flows have non-empty interiors for a broad class of functions and metrics.

Contribution

It establishes the non-empty interior property of these spectra for generic metrics and functions on negatively curved surfaces.

Findings

Lagrange and Markov spectra have non-empty interior for typical metrics.

Results apply to a large class of real functions on the unit tangent bundle.

The work extends understanding of dynamical spectra in negative curvature settings.

Abstract

We consider the Lagrange and the Markov dynamical spectra associated to a geodesic flow on a surface of negative curvature. We show that for a large set of real functions on the unit tangent bundle and for typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interior.