Ergodicity of the geodesic flow for groups with a contracting element
Rémi Coulon
TL;DR
The paper develops a dynamical framework for the geodesic flow of a proper metric space $X$ under a proper action by a group $G$ that contains a contracting element, extending the Hopf-Tsuji-Sullivan dichotomy beyond classical negatively curved manifolds. It introduces the coarse unit tangent bundle $SX$ and a Bowen–Margulis measure built from Patterson–Sullivan densities, then remainder-measures the flow on a suitably chosen quotient $(SX, rak L_G,m_G)$ by $G$. A central innovation is the reduced horoboundary $oundary X$ and a visual, ultrametric structure on a recurrent subset $ Lambda$, which allows a Hopf-type ergodic argument despite potential non-uniqueness of geodesics and non-proper action. The main result asserts that, under contracting-element hypotheses, the geodesic flow satisfies a dichotomy: either dissipative behavior in the convergent Poincaré-series case, or conservative and ergodic dynamics in the divergent case, with the radial limit set playing a decisive role and ergodicity extending to reduced boundary and related spaces. The strategy blends classical Hopf arguments with boundary-measurable decompositions and a careful construction of admissible sequences and cylinders, yielding broad ergodic conclusions in diverse geometric contexts such as CAT(-1) spaces, Teichmüller spaces, and properly convex Hilbert geometries.
Abstract
In this article we investigate the dynamical properties of the geodesic flow for a proper metric space endowed with a proper action by isometries of a group with a contracting element. We show that the existence of a contracting isometry is a sufficient evidence of negative curvature to carry in this context various results borrowed from hyperbolic geometry. In particular, we extend the so-called Hopf-Tsuji-Sullivan dichotomy proving that the geodesic flow is either conservative and ergodic or dissipative.