Decay of correlations for normally hyperbolic trapping

TL;DR

This paper proves exponential decay of correlations for evolution problems with normally hyperbolic trapping, using resonance theory and bounds on Green's functions, applicable to various physical and geometric flows.

Contribution

It establishes exponential decay of correlations for a broad class of hyperbolic flows with normally hyperbolic trapping, linking it to resonance-free regions and resolvent bounds.

Findings

Exponential decay of correlations proven for normally hyperbolic trapping.

Resonance-free strips are key to decay estimates.

Applicable to contact Anosov flows, molecular dynamics, and black hole geodesics.

Abstract

We prove that for evolution problems with normally hyperbolic trapping in phase space, correlations decay exponentially in time. Normal hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it. Flows with this structure include contact Anosov flows, classical flows in molecular dynamics, and null geodesic flows for black holes metrics. The decay of correlations is a consequence of the existence of resonance free strips for Green's functions (cut-off resolvents) and polynomial bounds on the growth sof those functions in the semiclassical parameter.