Decay of correlations for normally hyperbolic trapping

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 [21],[37], classi-cal flows in molecular dynamics [25],[26], and null geodesic flows for black holes metrics [17],[18],[42]. 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 of those functions in the semiclassical parameter. 1. Statement of results 1.1. Introduction. We prove the existence of resonance free strips for general semiclassical problems with normally hyperbolic trapped sets. The width of the strip is related to certain Lyapunov exponents and, for the spectral parameter in that strip, the Green’s function (cut-off resolvent) is polynomially bounded. Such estimates are closely related to exponential decay of correlations in classical dynamics and in scattering problems. The framework t