Ergodic geometry for non-elementary rank one manifolds

TL;DR

This paper extends classical ergodic theory results for negatively curved manifolds to rank one orbifolds, linking geodesic flow behavior with Poincaré series and analyzing conformal densities on limit sets.

Contribution

It generalizes the Hopf-Tsuji-Sullivan theorem to rank one orbifolds and studies properties of conformal densities supported on limit sets.

Findings

Extended ergodic theory results to rank one orbifolds.

Established connections between geodesic flow and Poincaré series.

Analyzed properties of conformal densities on limit sets.

Abstract

Let $X$ be a Hadamard manifold, and $\Gamma$ a non-elementary discrete group of isometries of $X$ which contains a rank one isometry. We relate the ergodic theory of the geodesic flow of the quotient orbifold $M=X/\Gamma$ to the behavior of the Poincar{\'e} series of $\Gamma$. Precisely, the aim of this paper is to extend the so-called theorem of Hopf-Tsuji-Sullivan -- well-known for manifolds of pinched negative curvature -- to the framework of rank one orbifolds. Moreover, we derive some important properties for $\Gamma$-invariant conformal densities supported on the geometric limit set of $\Gamma$.