Entropy rigidity and flexibility for suspension flows over Anosov diffeomorphisms

TL;DR

This paper investigates the conditions under which suspension flows over surface Anosov diffeomorphisms are conjugate to simpler models and explores the range of entropy values achievable, contributing to entropy rigidity and flexibility theory.

Contribution

It characterizes when suspension flows are conjugate to constant-time flows and demonstrates the full range of entropy values for these systems.

Findings

Suspension flow conjugacy occurs iff volume measure has maximal entropy.

The entropy with respect to volume and topological entropy can attain all values.

Results support entropy rigidity and flexibility programs for Anosov systems.

Abstract

For any $C^\infty$, area-preserving Anosov diffeomorphism $f$ of a surface, we show that a suspension flow over $f$ is $C^\infty$-conjugate to a constant-time suspension flow of a hyperbolic automorphism of the two torus if and only if the volume measure is the measure with maximal entropy. We also show that the the metric entropy with respect to the volume measure and the topological entropy of suspension flow over Anosov diffeomorphisms on torus achieve all possible values. Our results fit into two programs related to entropy rigidity and flexibility of Anosov systems.