A remark on rapid mixing for hyperbolic flows

TL;DR

This paper introduces a new criterion for rapid mixing in hyperbolic flows that relies on Diophantine conditions, broadening the scope of applicable systems beyond previous dimension-based criteria.

Contribution

It weakens the existing conditions for rapid mixing by replacing the positive box dimension requirement with a Diophantine ratio condition, enabling analysis of more systems.

Findings

New criterion for rapid mixing based on Diophantine ratios

Applicable to explicit examples previously inaccessible due to strict dimension conditions

Demonstrates broader applicability of hyperbolic flow mixing analysis

Abstract

We establish an improved criterion for rapid mixing of hyperbolic flows by weakening the requirement on the temporal distance function from positive box dimension to the existence of two values whose ratio is Diophantine. We also demonstrate the applicability of our results through explicit examples where the previous dimension condition were either too restrictive or computationally infeasible to verify.