Weak mixing suspension flows over shifts of finite type are universal

Anthony Quas; Terry Soo

doi:10.3934/jmd.2012.6.427

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2012, Volume 6, Issue 4: 427-449. Doi: 10.3934/jmd.2012.6.427

This issue Previous Article Next Article An algebraic characterization of expanding Thurston maps

Weak mixing suspension flows over shifts of finite type are universal

Anthony Quas^1, and
Terry Soo^2,

1.
Department of Mathematics and Statistics, University of Victoria, P.O. Box 3060 STN CSC, Victoria, B.C., V8W 3R4
2.
Department of Mathematics and Statistics, University of Victoria, PO BOX 3060 STN CSC, Victoria, BC V8W 3R4

Received: October 2011

Revised: July 2012

Published: December 2012

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Abstract

Let $S$ be an ergodic measure-preserving automorphism on a nonatomic probability space, and let $T$ be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Hölder ceiling function. We show that if the measure-theoretic entropy of $S$ is strictly less than the topological entropy of $T$, then there exists an embedding of the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen [5] and Ratner [31], we also obtain an embedding of the measure-preserving automorphism into a geodesic flow whenever the measure-theoretic entropy of $S$ is strictly less than the topological entropy of the time-one map of the geodesic flow.

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Mathematics Subject Classification: Primary: 37A35; Secondary: 37D40.

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