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

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.