Stretched-exponential mixing for surface semiflows and Anosov flows

TL;DR

This paper proves that certain surface semiflows and Anosov flows exhibit stretched-exponential mixing under specific regularity and non-cohomology conditions, extending previous results to more general hyperbolic systems.

Contribution

It establishes stretched-exponential mixing for surface semiflows and Anosov flows with Lipschitz stable foliations, generalizing known mixing rates to broader hyperbolic dynamics.

Findings

Surface semiflows with Lipschitz roof functions are stretched-exponentially mixing.

Anosov flows with Lipschitz stable foliations exhibit stretched-exponential mixing.

Results extend mixing properties to hyperbolic skew-product semiflows and attractors.

Abstract

For a surface semiflow that is a suspension of a \( C^{1+\alpha} \) expanding Markov interval map, we prove that, under the assumptions that the roof function is Lipschitz continuous and not cohomologous to a locally constant function, the semiflow exhibits stretched-exponential mixing with respect to the SRB measure. This result extends to hyperbolic skew-product semiflows and hyperbolic attractors. Specifically, codimension-one topologically mixing Anosov flows with Lipschitz continuous stable foliations demonstrate stretched-exponential mixing with respect to their SRB measures.