Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

A Functional Approach

  • Viviane Baladi

Table of contents

  1. Front Matter
    Pages I-XV
  2. Viviane Baladi
    Pages 1-17
  3. Smooth expanding maps

    1. Front Matter
      Pages 19-20
    2. Viviane Baladi
      Pages 79-119
  4. Smooth hyperbolic maps

    1. Front Matter
      Pages 121-122
    2. Viviane Baladi
      Pages 123-155
    3. Viviane Baladi
      Pages 209-234
  5. Back Matter
    Pages 235-291

About this book


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.


dynamical zeta functions Ruelle transfer operators Anosov diffeomorphisms anisotropic Banach Spaces linear response Gibbs states and equilibrium states dynamical determinants MSC (2010): 37C30, 37D20, 37D35

Authors and affiliations

  • Viviane Baladi
    • 1
  1. 1.IMJ-PRGSorbonne Université and CNRSParisFrance

Bibliographic information

  • DOI
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-77660-6
  • Online ISBN 978-3-319-77661-3
  • Series Print ISSN 0071-1136
  • Series Online ISSN 2197-5655
  • Buy this book on publisher's site