Equidistribution results for geodesic flows

TL;DR

This paper investigates how geodesic flows distribute over a closed manifold, establishing large deviations bounds and approximation methods for equilibrium states using advanced dynamical systems theory.

Contribution

It extends the understanding of geodesic flow distribution by proving large deviations bounds and a contraction principle, enabling approximation of equilibrium states for continuous potentials.

Findings

Established large deviations lower and upper bounds.

Proved a contraction principle for geodesic flow.

Provided a method to approximate equilibrium states.

Abstract

Using the works of Ma\~n\'e \cite{Ma} and Paternain \cite{Pat} we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal{C}^{\infty}$ Riemannian metric. We prove large deviations lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.