Pollicott-Ruelle resonances for open systems

TL;DR

This paper introduces Pollicott-Ruelle resonances for open hyperbolic systems, extending the understanding of decay of correlations and meromorphic continuation of the Ruelle zeta function on noncompact manifolds.

Contribution

It defines and analyzes Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic manifolds and general open hyperbolic systems, extending prior theories.

Findings

Resonances are poles of the meromorphically continued resolvent.

Ruelle zeta function extends meromorphically to the entire complex plane.

Decay of classical correlations is characterized by these resonances.

Abstract

We define Pollicott-Ruelle resonances for geodesic flows on noncompact asymptotically hyperbolic negatively curved manifolds, as well as for more general open hyperbolic systems related to Axiom A flows. These resonances are the poles of the meromorphic continuation of the resolvent of the generator of the flow and they describe decay of classical correlations. As an application, we show that the Ruelle zeta function extends meromorphically to the entire complex plane.