Spectra of differentiable hyperbolic maps

TL;DR

This paper investigates the spectral properties of transfer operators in smooth hyperbolic dynamics, establishing new links with dynamical determinants and providing accessible explanations and proofs for key results in the field.

Contribution

It introduces novel results connecting spectra with dynamical determinants and offers simplified, reader-friendly proofs of fundamental theorems, including Ruelle's result on expanding endomorphisms.

Findings

Spectral properties of transfer operators are related to dynamical determinants.

New proofs of classical results in hyperbolic dynamics are provided.

Enhanced understanding of spectra in smooth hyperbolic maps.

Abstract

This note is about the spectral properties of transfer operators associated to smooth hyperbolic dynamics. In the first two sections (written in 2006), we state our new results relating such spectra with dynamical determinants, first announced at the conference ``Traces in Geometry, Number Theory and Quantum Fields" at the Max Planck Institute, Bonn, October 2005. In the last two sections, we give a reader-friendly presentation of some key ideas in our work in the simplest possible settings, including a new proof of a result of Ruelle on expanding endomorphisms. (These last two sections are a revised version of the lecture notes given during the workshop ``Resonances and Periodic Orbits: Spectrum and Zeta functions in Quantum and Classical Chaos" at Institut Henri Poincar\'e, Paris, July 2005.) (Revised version, submitted for publication)