Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

TL;DR

This paper introduces a hierarchy of fractal Weyl laws for resonance states in open chaotic systems with mixed phase space, revealing how different regions support localized resonance states.

Contribution

It extends fractal Weyl law concepts to mixed phase spaces, establishing a hierarchy based on the hierarchical structure of chaotic regions.

Findings

Hierarchical resonance states localize on different phase-space regions

Numerical verification using standard map and hierarchical model

Multiple fractal Weyl laws corresponding to phase-space hierarchy

Abstract

In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.