Overview
- 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
Part of the book series: Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics (MATHE3, volume 68)
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Table of contents (7 chapters)
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Front Matter
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Smooth expanding maps
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Front Matter
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Back Matter
Keywords
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.
Reviews
“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)
Authors and Affiliations
About the author
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
DOI: https://doi.org/10.1007/978-3-319-77661-3
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing AG, part of Springer Nature 2018
Hardcover ISBN: 978-3-319-77660-6Published: 28 May 2018
Softcover ISBN: 978-3-030-08505-6Published: 09 February 2019
eBook ISBN: 978-3-319-77661-3Published: 09 May 2018
Series ISSN: 0071-1136
Series E-ISSN: 2197-5655
Edition Number: 1
Number of Pages: XV, 291
Number of Illustrations: 1 b/w illustrations
Topics: Dynamical Systems and Ergodic Theory, Functional Analysis, Operator Theory