Resonances in hyperbolic dynamics

TL;DR

This paper investigates the distribution of resonances in hyperbolic dynamical systems related to wave propagation, deriving bounds and criteria for resonance gaps using semiclassical and dynamical systems techniques.

Contribution

It introduces fractal Weyl bounds for resonance counting and provides dynamical criteria for resonance gaps in hyperbolic and normally hyperbolic trapped sets.

Findings

Established fractal Weyl upper bounds for resonances.

Derived dynamical criteria for the existence of resonance gaps.

Analyzed cases with normally hyperbolic trapped sets.

Abstract

The study of wave propagation outside bounded obstacles uncovers the existence of resonances for the Laplace operator, which are complex-valued generalized eigenvalues, relevant to estimate the long time asymptotics of the wave. In order to understand distribution of these resonances at high frequency, we employ semiclassical tools, which leads to considering the classical scattering problem, and in particular the set of trapped trajectories. We focus on "chaotic" situations, where this set is a hyperbolic repeller, generally with a fractal geometry. In this context, we derive fractal Weyl upper bounds for the resonance counting; we also obtain dynamical criteria ensuring the presence of a resonance gap. We also address situations where the trapped set is a normally hyperbolic submanifold, a case which can help analyzing the long time properties of (classical) Anosov contact flows through semiclassical methods.