Explicit examples of resonances for Anosov maps of the torus

TL;DR

This paper introduces new anisotropic Hilbert spaces to analyze resonances of Anosov maps on the torus, simplifying previous methods and enabling the construction of additional examples.

Contribution

It develops alternative anisotropic Hilbert spaces that streamline the analysis of resonances for Anosov diffeomorphisms on the torus, expanding the class of constructible examples.

Findings

Simplified the analysis of resonances using new Hilbert space constructions

Enabled the creation of additional explicit examples of Anosov resonances

Provided a framework for further exploration of dynamical resonances

Abstract

In [23], Slipantschuk, Bandtlow and Just gave concrete examples of Anosov diffeomorphisms of the two-torus for which their resonances could be completely described. Their approach was based on composition operators acting on analytic anisotropic Hilbert spaces, and in this note we present a construction of alternative anisotropic Hilbert spaces which helps to simplify parts of their analysis and gives scope for constructing further examples.