Generic non-trivial resonances for Anosov diffeomorphisms

TL;DR

This paper investigates how small real analytic perturbations affect hyperbolic automorphisms on the 2-torus, revealing the generic presence of non-trivial Ruelle resonances and their relation to volume preservation.

Contribution

It demonstrates the generic existence of non-trivial Ruelle resonances for perturbed automorphisms and analyzes volume-preserving versus non-preserving cases.

Findings

Ruelle resonances are generically present after perturbations.

Some perturbations preserve volume, others do not.

The Koopman and transfer operators are nuclear of order 0 on suitable spaces.

Abstract

We study real analytic perturbations of hyperbolic linear automorphisms on the 2-torus. The Koopman and the transfer operator are nuclear of order 0 when acting on a suitable Hilbert space. We show the generic existence of non-trivial Ruelle resonances for both operators. We prove that some of the perturbations preserve the volume and some of them do not.