On the Lagrange and Markov Dynamical Spectra for Anosov Flows in dimension 3

TL;DR

This paper demonstrates that for most conservative Anosov flows in three dimensions, the associated Lagrange and Markov spectra contain intervals, indicating rich dynamical complexity.

Contribution

It establishes the generic presence of non-empty interiors in the Lagrange and Markov spectra for a broad class of 3D Anosov flows, including geodesic flows of negative curvature.

Findings

Both spectra have non-empty interior for typical flows.

Results apply to geodesic flows of negative curvature.

Spectra exhibit complex dynamical behavior.

Abstract

We consider the Lagrange and the Markov dynamical spectra associated with a conservative Anosov flow on a compact manifold of dimension $3$ (including geodesic flows of negative curvature and suspension flows). We show that for a large set of real functions and typical conservative Anosov flows, both the Lagrange and Markov dynamical spectra have a non-empty interior.