Weak mixing suspension flows over shifts of finite type are universal

TL;DR

This paper demonstrates that weak mixing suspension flows over shifts of finite type are capable of embedding any ergodic measure-preserving automorphism with lower entropy, highlighting their universality in dynamical systems.

Contribution

It establishes the universality of weak mixing suspension flows over shifts of finite type for embedding automorphisms with lower entropy, extending symbolic dynamics to geodesic flows.

Findings

Embedding exists when measure-theoretic entropy is less than topological entropy.

Embeddings apply to geodesic flows on negatively curved surfaces.

Shows universality of certain suspension flows in dynamical systems.

Abstract

Let S be an ergodic measure-preserving automorphism on a non-atomic 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 Holder 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 from 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 and Ratner, we also obtain an embedding from 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.