Ergodic geometry for non-elementary rank one manifolds Article Swipe

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Ergodic theory Quotient Mathematics Pure mathematics Orbifold Rank (graph theory) Invariant (physics) Discrete group Geodesic Isometry (Riemannian geometry) Geodesic flow Conformal map Curvature Ricci-flat manifold Scalar curvature Combinatorics Mathematical analysis Physics Group (periodic table) Geometry Mathematical physics Quantum mechanics

Jean-Claude Picaud , Gabriele Link ·

YOU? · · 2016 · Open Access · · DOI: https://doi.org/10.3934/dcds.2016072 · OA: W2963987731

Let $X$ be a Hadamard manifold, and $\Gamma\subset Is(X)$ a non-elementary discrete subgroup of isometries of $X$ whichcontains 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 Poincaré 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 geometriclimit set of $\Gamma$.