Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

A Functional Approach

verfasst von: Prof. Dr. Viviane Baladi

Verlag: Springer International Publishing

Buchreihe : Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The spectra of transfer operators associated to dynamical systems, when acting on suitable Banach spaces, contain key information about the ergodic properties of the systems. Focusing on expanding and hyperbolic maps, this book gives a self-contained account on the relation between zeroes of dynamical determinants, poles of dynamical zeta functions, and the discrete spectra of the transfer operators.

In the hyperbolic case, the first key step consists in constructing a suitable Banach space of anisotropic distributions. The first part of the book is devoted to the easier case of expanding endomorphisms, showing how the (isotropic) function spaces relevant there can be studied via Paley–Littlewood decompositions, and allowing easier access to the construction of the anisotropic spaces which is performed in the second part.

This is the first book describing the use of anisotropic spaces in dynamics. Aimed at researchers and graduate students, it presents results and techniques developed since the beginning of the twenty-first century.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this chapter, we first define the transfer operator, dynamical determinants, and the spectral and determinantal resonances of a weighted differentiable dynamical system. We then state the main results linking these resonances. We also discuss anisotropic spaces and the techniques used to prove these results, illustrating them by simple examples.

Viviane Baladi

Smooth expanding maps

Frontmatter

Chapter 2. Smooth expanding maps: The spectrum of the transfer operator

Abstract

This chapter presents a variant of Ruelle’s bound on the essential spectral radius of transfer operators associated with differentiable expanding dynamics and weights, replacing the Hölder spaces by Sobolev spaces. The chapter ends with the Gouëzel-Keller-Liverani perturbation theory, which will also be applicable in the hyperbolic setting of Part II.

Viviane Baladi

Chapter 3. Smooth expanding maps: Dynamical determinants

Abstract

The main result of this chapter is a variant of Ruelle’s theorem on the dynamical determinants of transfer operators associated with differentiable expanding dynamics and weights. The proof uses the Milnor-Thurston kneading operator approach. The contents of this chapter are a blueprint for the technically more involved situation of hyperbolic dynamics and the corresponding anisotropic Banach spaces in Part II.

Viviane Baladi

Smooth hyperbolic maps

Frontmatter

Chapter 4. Anisotropic Banach spaces defined via cones

Abstract

The main result of this chapter is a bound on the essential spectral radius of a weighted transfer operator associated with a differentiable diffeomorphism on a hyperbolic basic set and a differentiable weight function. The operator acts on two scales of anisotropic spaces of distributions on the manifold defined using cones (in the cotangent space) adapted to the diffeomorphism.

Viviane Baladi

Chapter 5. A variational formula for the essential spectral radius

Abstract

This chapter describes a third scale of anisotropic Banach spaces of distributions, for which the best known bounds on the essential spectral radius of the transfer operator are known, improving those given in Chapter 4. The last section implements the Gouëzel-Keller-Liverani perturbation theory for this third type of Banach spaces.

Viviane Baladi

Chapter 6. Dynamical determinants for smooth hyperbolic dynamics

Abstract

As in Chapters 4 and 5, we consider a diffeomorphism on a hyperbolic basic set and a differentiable weight. In this chapter, we study the associated weighted dynamical determinant, giving a lower bound on the disc in which this determinant is analytic and where its zeroes admit a spectral interpretation. We apply the results obtained on the weighted dynamical determinant to study the dynamical zeta function.

Viviane Baladi

Chapter 7. Two applications of anisotropic spaces

Abstract

In this chapter, we first interpret the maximal eigenvectors of positively weighted transfer operators as equilibrium measures, or (for attractors) as SRB measures, in the spirit of the work of Gouëzel-Liverani, recovering classical results of existence, uniqueness, and exponential mixing. Then we present Tsujii’s unpublished proof of Anosov’s theorem using anisotropic spaces.

Viviane Baladi

Backmatter

Titel: Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps
verfasst von: Prof. Dr. Viviane Baladi
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-77661-3
Print ISBN: 978-3-319-77660-6
DOI: https://doi.org/10.1007/978-3-319-77661-3