Resonances near the real axis for manifolds with hyperbolic trapped sets

TL;DR

This paper investigates the distribution of resonances near the real axis for certain manifolds with hyperbolic trapped sets, revealing sub-linear growth in resonance counting and linking scattering poles to the trapped set's Hausdorff dimension.

Contribution

It establishes conditions under which resonance counting functions grow sub-linearly and connects scattering poles to the geometric structure of the trapped set.

Findings

Resonance counting functions grow sub-linearly under hyperbolicity assumptions.

Scattering poles provide information about the Hausdorff dimension of the trapped set.

Results apply to manifolds Euclidean at infinity with hyperbolic trapped sets.

Abstract

For manifolds Euclidian at infinity and compact perturbations of the Laplacian, we show that under assumptions involving hyperbolicity of the classical flow on the trapped set and its period spectrum, there are strips below the real axis where the resonance counting function grows sub-linearly. We also provide an inverse result, showing that the knowledge of the scattering poles can give some information about the Hausdorff dimension of the trapped set when the classical flow satisfies the Axiom-A condition.