# Hopf-Tsuji-Sullivan dichotomy for quotients of Hadamard spaces with a rank one isometry

## TL;DR

This paper extends the Hopf-Tsuji-Sullivan dichotomy to the geodesic flow on quotients of proper Hadamard spaces with rank one isometries, using Ricks' measure, generalizing previous manifold results.

## Contribution

It proves the Hopf-Tsuji-Sullivan dichotomy for geodesic flows in a broader non-manifold setting with respect to Ricks' measure, generalizing prior work.

## Key findings

- Establishes dichotomy for geodesic flow on Hadamard space quotients.
- Uses Ricks' measure to extend previous results.
- Generalizes manifold results to non-manifold Hadamard spaces.

## Abstract

Let $X$ be a proper Hadamard space and $\Gamma< Isom(X)$ a non-elementary discrete group of isometries with a rank one isometry. We discuss and prove Hopf-Tsuji-Sullivan dichotomy for the geodesic flow on the set of parametrized geodesics of the quotient of $X$ by $\Gamma$ and with respect to Ricks' measure introduced in [MR3628926]. This generalizes previous work of the author and J. C. Picaud on Hopf-Tsuji-Sullivan dichotomy in the analogous manifold setting and with respect to Knieper's measure.