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

Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps

A Functional Approach

Authors: Baladi, Viviane

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  • Collects recent results on transfer operators, anisotropic Banach spaces, and dynamical determinants of hyperbolic systems
  • Gives a self-contained account of proofs (some of them new) starting with the basic case of expanding maps for easier readability
  • Each chapter ends with a list of open research problems
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  • ISBN 978-3-319-77661-3
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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.

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.

Table of contents (7 chapters)

Table of contents (7 chapters)
  • Introduction

    Pages 1-17

    Baladi, Viviane

  • Smooth expanding maps: The spectrum of the transfer operator

    Pages 21-77

    Baladi, Viviane

  • Smooth expanding maps: Dynamical determinants

    Pages 79-119

    Baladi, Viviane

  • Anisotropic Banach spaces defined via cones

    Pages 123-155

    Baladi, Viviane

  • A variational formula for the essential spectral radius

    Pages 157-182

    Baladi, Viviane

Buy this book

eBook $69.99
$99.00 (listprice)
price for USA in USD (gross)
valid through July 31, 2020
  • ISBN 978-3-319-77661-3
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $89.99
$129.00 (listprice)
price for USA in USD
valid through July 31, 2020
  • ISBN 978-3-319-77660-6
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
Softcover $89.99
$129.00 (listprice)
price for USA in USD
valid through July 31, 2020
  • ISBN 978-3-030-08505-6
  • Free shipping for individuals worldwide
  • Immediate ebook access, if available*, with your print order
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps
Book Subtitle
A Functional Approach
Authors
Series Title
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
Series Volume
68
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing AG, part of Springer Nature
eBook ISBN
978-3-319-77661-3
DOI
10.1007/978-3-319-77661-3
Hardcover ISBN
978-3-319-77660-6
Softcover ISBN
978-3-030-08505-6
Series ISSN
0071-1136
Edition Number
1
Number of Pages
XV, 291
Number of Illustrations
1 b/w illustrations
Topics

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