© 2018

Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

A Functional Approach


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

  1. 1.IMJ-PRGSorbonne Université and CNRSParisFrance

About the authors

Viviane Baladi has been working as a researcher for CNRS since 1990 (currently Directeur de Recherche at Institut de Mathématiques de Jussieu-Paris Rive Gauche), spending several academic years on leave to teach at the Universities of Geneva and Copenhagen, and at the Eidgenössische Technische Hochschule Zürich. Her interest in dynamical zeta functions and transfer operators developed during her Ph.D. in Geneva. She has since applied transfer operators to algorithmics, linear response and the violation thereof, and rates of mixing for Sinai billiards. She has further played a key role introducing anisotropic spaces of distributions in dynamical systems.

Bibliographic information

  • Book Title Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps
  • Book Subtitle A Functional Approach
  • Authors Viviane Baladi
  • Series Title Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
  • Series Abbreviated Title Ergebnisse Mathematik 3.F.
  • DOI
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-77660-6
  • Softcover ISBN 978-3-030-08505-6
  • eBook ISBN 978-3-319-77661-3
  • Series ISSN 0071-1136
  • Series E-ISSN 2197-5655
  • Edition Number 1
  • Number of Pages XV, 291
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Dynamical Systems and Ergodic Theory
    Functional Analysis
    Operator Theory
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“I highly recommend this book for graduate students, researchers interested in the modern developments of dynamical systems theory and quantum chaos. This Fourier analytic approach has already had a deep impact on the subject and is now used widely in other related fields.” (Frédéric Naud, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 122, 2020)