Dynamical zeta functions and dynamical determinants for hyperbolic maps : a functional approach
 Responsibility
 Viviane Baladi.
 Publication
 Cham, Switzerland : Springer, [2018]
 Physical description
 xv, 291 pages ; 25 cm.
 Series
 Ergebnisse der Mathematik und ihrer Grenzgebiete ; 3. Folge, Bd. 68.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA351 .B35 2018  Unknown 
More options
Creators/Contributors
 Author/Creator
 Baladi, Viviane, author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 281288) and index.
 Contents

 Introduction
 Statistical properties of chaotic differentiable dynamical systems
 Transfer operators : Dynamical determinants : Resonances
 Main results : examples
 Main techniques
 Comments
 Smooth expanding maps
 Smooth expanding maps : the spectrum of the transfer operator
 Transfer operators for smooth expanding maps on Holder functions
 Transfer operators for smooth expanding maps on Sobolev spaces
 Isotropic Sobolev spaces H... and good systems of charts
 Bounding the essential spectral radius (Theorem 2.15)
 The key local Lasota Yorke bound (Lemma 2.21)
 Fragmentation and reconstitution : technical lemmas
 The essential spectral radius on Sobolev spaces : interpolation
 Complex interpolation
 Proof of Theorem 2.15 on ... for integer differentiability
 The essential spectral radius : dyadic decomposition
 A PaleyLittlewood description of H... and C...
 Proof of Lemma 2.21 and Theorem 2.15 : the general case
 Spectral stability and linear response à la GouëzelKellerLiverani
 Problems
 Comments
 Smooth expanding maps : dynamical determinants
 Ruelle's theorem on the dynamical determinant
 Dynamical zeta functions
 Ruelle's theorem via kneading determinants
 Outline
 Flat traces
 Dynamical determinants : completing the proof of Theorem 3.5
 Proof of Theorem 3.5 if ... d + t
 Nuclear power decomposition via approximation numbers
 Asymptotic vanishing of flat traces of the noncompact term
 The case ... d + t of low differentiability
 Problems
 Comments
 Smooth hyperbolic maps
 Anisotropic Banach spaces defined via cones
 Transfer operators for hyperbolic dynamics
 Hyperbolic dynamics and anisotropic spaces
 Bounding the essential spectral radius (Theorem 4.6)
 Reducing to the transitive case
 The spaces ... and ...
 Charts and cone systems adapted to (T, V)
 Formal definition of the spaces ... and ...
 The local Lasota Yorke lemma and the proof of Theorem 4.6
 The PaleyLittlewood description of the spaces and the local LasotaYorke lemma
 Fragmentation, reconstitution, and the proof of Theorem 4.6
 Problems
 Comments
 A variational formula for the essential spectral radius
 Yet another anisotropic Banach space : ...
 Defining ...
 Bounding the essential spectral radius on Bts (Theorem 5.1)
 Spectral stability and linear response
 Problems
 Comments
 Dynamical determinants for smooth hyperbolic dynamics
 Dynamical determinants via regularised determinants and flat traces
 Proof of Theorem 6.2 on ...
 Theorem 6.2 in low differentiability ...
 Operators on vector bundles and dynamical zeta functions
 Problems
 Comments
 Two applications of anisotropic spaces
 Equilibrium measures and SRB measures
 Peripheral spectrum and equilibrium measures
 Grassmannians and the measure of maximal entropy
 SRB measures for hyperbolic attractors
 Tsujii's proof of Anosov's theorem
 Problems
 Comments
 Appendices
 Spectral theory
 Bounding the essential spectral radius : Hennion's theorem
 Eigenvalues and eigenvectors for different Banach spaces
 An abstract perturbation result of GouëzelKellerLiverani
 Nuclear operators and approximation numbers
 Thermodynamic formalism : nonmultiplicative topological pressure
 Properly supported operators (pseudolocality)
 Alternative proofs for C... dynamics and weights
 Elements of symbolic calculus
 Essential spectral radius for C... expanding maps
 Dynamical determinants for C... expanding maps
 The essential spectral radius for C... hyperbolic maps
 References
 Index.
 Publisher's Summary
 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 selfcontained 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 PaleyLittlewood 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 twentyfirst century.
(source: Nielsen Book Data)9783319776606 20180917
Subjects
Bibliographic information
 Publication date
 2018
 Series
 Ergebnisse der Mathematik und ihrer Grenzgebiete = A Series of modern surveys in mathematics, 00711136 ; 3. Folge, volume 68
 Note
 "ISSN 21975655 (electronic)"Title page verso.
 ISBN
 9783319776606 (hd.bd.)
 9783319776613 (online)
 3319776606