On the Growth of hyperbolic geodesics in rank 1 manifolds

TL;DR

This paper establishes a formula linking the topological pressure of geodesic flow in rank 1 manifolds to hyperbolic geodesic growth and proves an equidistribution result for these geodesics relative to equilibrium states.

Contribution

It generalizes Knieper's result to non-constant potentials, providing new insights into the distribution of hyperbolic geodesics in rank 1 manifolds.

Findings

Formula for topological pressure in terms of hyperbolic geodesic growth

Equidistribution of hyperbolic geodesics with respect to equilibrium states

Extension of Knieper's result to non-constant potentials

Abstract

We give a formula for the topological pressure of the geodesic flow of a compact rank 1 manifold in terms of the growth of the number of closed hyperbolic (rank 1) geodesics. We derive an equidistribution result for these geodesics with respect to equilibrium states. This generalize partially a result of G. Knieper \cite{kni} to non constant potentiels.