Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization

2024/04/15 by Dongryul M. Kim, Hee Oh, Kim, Dongryul M. +1 · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2404.09745

Abstract

For a geometrically finite Kleinian group Γ, the Bowen-Margulis-Sullivan measure is finite and is the unique measure of maximal entropy for the geodesic flow, as shown by Sullivan and Otal-Peigné respectively. Moreover, it is strongly mixing by a result of Babillot. We obtain a higher-rank analogue of this theorem. Given a relatively Anosov subgroup Γ of a semisimple real algebraic group, there is a family of flow spaces parameterized by linear forms tangent to the growth indicator. We construct a reparameterization of each flow space by the geodesic flow on the Groves-Manning space of Γ which exhibits exponential expansion along unstable foliations. Using this reparameterization, we prove that the Bowen-Margulis-Sullivan measure of each flow space is finite and is the unique measure of maximal entropy. Moreover, it is strongly mixing.

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• Classification of horospherical invariant measures in higher rank
• Kleinian viewpoints on higher rank worlds

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