Exponential Mixing for Hyperbolic Flows on Non-Compact Spaces

TL;DR

This paper establishes exponential decay of correlations for a family of hyperbolic flows on non-compact spaces, including the geodesic flow on the modular surface, using a suspension flow approach and Dolgopyat's method.

Contribution

It introduces a new suspension model with a countable Markov partition for hyperbolic flows on non-compact spaces, extending exponential mixing results.

Findings

Proves exponential decay of correlations for these flows.

Constructs a suspension model with a hyperbolic Poincaré map.

Recovers Ratner's exponential mixing result for the modular surface.

Abstract

We introduce a family of hyperbolic flows on non-compact phase spaces that includes the geodesic flow on the modular surface. For these systems we prove exponential decay of correlations for sufficiently regular observables with respect to its SRB measure. Our approach follows the dynamical method of Dolgopyat and subsequent developments for suspension flows with uniformly hyperbolic Poincar\'e maps satisfying a uniform non-integrability condition. To fit this framework, we construct a suspension model via a triple inducing scheme that yields a uniformly hyperbolic Poincar\'e map with a countable Markov partition. We show that the resulting roof function is cohomologous to one that is constant along stable leaves and satisfies the required non-integrability and tail conditions. As an application, we recover a dynamical proof on Ratner's exponential mixing for the geodesic flow on the modular surface.