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Topological flows for hyperbolic groups

Part of: Special aspects of infinite or finite groups Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  30 October 2020

RYOKICHI TANAKA Open the ORCID record for RYOKICHI TANAKA [Opens in a new window]
RYOKICHI TANAKA*
Affiliation:
Mathematical Institute, Tohoku University, Sendai980-8578, Japan
*
e-mail: [email protected]
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Abstract

We show that for every non-elementary hyperbolic group the Bowen–Margulis current associated with a strongly hyperbolic metric forms a unique group-invariant Radon measure class of maximal Hausdorff dimension on the boundary square. Applications include a characterization of roughly similar hyperbolic metrics via mean distortion.

Keywords

hyperbolic groupgeodesic currentPatterson–Sullivan measureCannon’s automatic structureHausdorff dimension

MSC classification

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 37D35: Thermodynamic formalism, variational principles, equilibrium states 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3474 - 3520
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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