The Resource Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book)
Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book)
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The item Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.This item is available to borrow from 1 library branch.
Resource Information
The item Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Liverpool.
This item is available to borrow from 1 library branch.
 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 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 twentyfirst century
 Language
 eng
 Extent
 1 online resource (xv, 291 pages)
 Contents

 Intro; Preface; Contents; Index of notations; 1 Introduction; 1.1 Statistical properties of chaotic differentiable dynamical systems; 1.2 Transfer operators. Dynamical determinants. Resonances; 1.3 Main results. Examples; 1.4 Main techniques; Comments; Part I Smooth expanding maps; 2 Smooth expanding maps: The spectrum of the transfer operator; 2.1 Transfer operators for smooth expanding maps on Hölder functions; 2.2 Transfer operators for smooth expanding maps on Sobolev spaces; 2.2.1 Isotropic Sobolev spaces Htp and good systems of charts
 2.2.2 Bounding the essential spectral radius (Theorem 2.15)2.2.3 The key local LasotaYorke bound (Lemma 2.21); 2.2.4 Fragmentation and reconstitution: Technical lemmas; 2.3 The essential spectral radius on Sobolev spaces: Interpolation; 2.3.1 Complex interpolation; 2.3.2 Proof of Theorem 2.15 on Htp for integer differentiability; 2.4 The essential spectral radius: Dyadic decomposition; 2.4.1 A PaleyLittlewood description of Htp and Ct*; 2.4.2 Proof of Lemma 2.21 and Theorem 2.15: The general case; 2.5 Spectral stability and linear response à la GouëzelKellerLiverani; Problems; Comments
 3 Smooth expanding maps: Dynamical determinants3.1 Ruelle's theorem on the dynamical determinant; 3.1.1 Dynamical zeta functions; 3.2 Ruelle's theorem via kneading determinants; 3.2.1 Outline; 3.2.2 Flat traces; 3.3 Dynamical determinants: Completing the proof of Theorem 3.5; 3.3.1 Proof of Theorem 3.5 if >d+t; 3.3.2 Nuclear power decomposition via approximation numbers; 3.3.3 Asymptotic vanishing of flat traces of the noncompact term; 3.3.4 The case d+t of low differentiability; Problems; Comments; Part II Smooth hyperbolic maps; 4 Anisotropic Banach spaces defined via cones
 4.1 Transfer operators for hyperbolic dynamics4.1.1 Hyperbolic dynamics and anisotropic spaces; 4.1.2 Bounding the essential spectral radius (Theorem 4.6); 4.1.3 Reducing to the transitive case; 4.2 The spaces Wp,*t,s and Wp,**t,s; 4.2.1 Charts and cone systems adapted to (T,V); 4.2.2 Formal definition of the spaces Wp,*t,s and Wp,**t,s; 4.3 The local LasotaYorke lemma and the proof of Theorem 4.6; 4.3.1 The PaleyLittlewood description of the spaces and the local LasotaYorke lemma; 4.3.2 Fragmentation, reconstitution, and the proof of Theorem 4.6; Problems; Comments
 5 A variational formula for the essential spectral radius5.1 Yet another anisotropic Banach space: Bt,s; 5.1.1 Defining Bt,s; 5.2 Bounding the essential spectral radius on Bt,s (Theorem 5.1); 5.3 Spectral stability and linear response; Problems; Comments; 6 Dynamical determinants for smooth hyperbolic dynamics; 6.1 Dynamical determinants via regularised determinants and flat traces; 6.2 Proof of Theorem 6.2 on dT,g(z) if r1> d+ ts; 6.3 Theorem 6.2 in low differentiability r1d+ts; 6.4 Operators on vector bundles and dynamical zeta functions; Problems; Comments
 Isbn
 9783319776613
 Label
 Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach
 Title
 Dynamical Zeta functions and dynamical determinants for hyperbolic maps
 Title remainder
 a functional approach
 Statement of responsibility
 Viviane Baladi
 Language
 eng
 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 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 twentyfirst century
 Cataloging source
 GW5XE
 http://library.link/vocab/creatorName
 Baladi, Viviane
 Dewey number
 515/.56
 Illustrations
 illustrations
 Index
 index present
 LC call number
 QA351
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,
 Series volume
 volume 68
 http://library.link/vocab/subjectName

 Functions, Zeta
 Dynamics
 Banach spaces
 Geometry, Hyperbolic
 Label
 Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book)
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Intro; Preface; Contents; Index of notations; 1 Introduction; 1.1 Statistical properties of chaotic differentiable dynamical systems; 1.2 Transfer operators. Dynamical determinants. Resonances; 1.3 Main results. Examples; 1.4 Main techniques; Comments; Part I Smooth expanding maps; 2 Smooth expanding maps: The spectrum of the transfer operator; 2.1 Transfer operators for smooth expanding maps on Hölder functions; 2.2 Transfer operators for smooth expanding maps on Sobolev spaces; 2.2.1 Isotropic Sobolev spaces Htp and good systems of charts
 2.2.2 Bounding the essential spectral radius (Theorem 2.15)2.2.3 The key local LasotaYorke bound (Lemma 2.21); 2.2.4 Fragmentation and reconstitution: Technical lemmas; 2.3 The essential spectral radius on Sobolev spaces: Interpolation; 2.3.1 Complex interpolation; 2.3.2 Proof of Theorem 2.15 on Htp for integer differentiability; 2.4 The essential spectral radius: Dyadic decomposition; 2.4.1 A PaleyLittlewood description of Htp and Ct*; 2.4.2 Proof of Lemma 2.21 and Theorem 2.15: The general case; 2.5 Spectral stability and linear response à la GouëzelKellerLiverani; Problems; Comments
 3 Smooth expanding maps: Dynamical determinants3.1 Ruelle's theorem on the dynamical determinant; 3.1.1 Dynamical zeta functions; 3.2 Ruelle's theorem via kneading determinants; 3.2.1 Outline; 3.2.2 Flat traces; 3.3 Dynamical determinants: Completing the proof of Theorem 3.5; 3.3.1 Proof of Theorem 3.5 if >d+t; 3.3.2 Nuclear power decomposition via approximation numbers; 3.3.3 Asymptotic vanishing of flat traces of the noncompact term; 3.3.4 The case d+t of low differentiability; Problems; Comments; Part II Smooth hyperbolic maps; 4 Anisotropic Banach spaces defined via cones
 4.1 Transfer operators for hyperbolic dynamics4.1.1 Hyperbolic dynamics and anisotropic spaces; 4.1.2 Bounding the essential spectral radius (Theorem 4.6); 4.1.3 Reducing to the transitive case; 4.2 The spaces Wp,*t,s and Wp,**t,s; 4.2.1 Charts and cone systems adapted to (T,V); 4.2.2 Formal definition of the spaces Wp,*t,s and Wp,**t,s; 4.3 The local LasotaYorke lemma and the proof of Theorem 4.6; 4.3.1 The PaleyLittlewood description of the spaces and the local LasotaYorke lemma; 4.3.2 Fragmentation, reconstitution, and the proof of Theorem 4.6; Problems; Comments
 5 A variational formula for the essential spectral radius5.1 Yet another anisotropic Banach space: Bt,s; 5.1.1 Defining Bt,s; 5.2 Bounding the essential spectral radius on Bt,s (Theorem 5.1); 5.3 Spectral stability and linear response; Problems; Comments; 6 Dynamical determinants for smooth hyperbolic dynamics; 6.1 Dynamical determinants via regularised determinants and flat traces; 6.2 Proof of Theorem 6.2 on dT,g(z) if r1> d+ ts; 6.3 Theorem 6.2 in low differentiability r1d+ts; 6.4 Operators on vector bundles and dynamical zeta functions; Problems; Comments
 Extent
 1 online resource (xv, 291 pages)
 Form of item
 online
 Isbn
 9783319776613
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319776613
 Other physical details
 illustration.
 System control number

 on1035635786
 (OCoLC)1035635786
 Label
 Dynamical Zeta functions and dynamical determinants for hyperbolic maps : a functional approach, Viviane Baladi, (electronic book)
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Carrier MARC source
 rdacarrier
 Content category
 text
 Content type code

 txt
 Content type MARC source
 rdacontent
 Contents

 Intro; Preface; Contents; Index of notations; 1 Introduction; 1.1 Statistical properties of chaotic differentiable dynamical systems; 1.2 Transfer operators. Dynamical determinants. Resonances; 1.3 Main results. Examples; 1.4 Main techniques; Comments; Part I Smooth expanding maps; 2 Smooth expanding maps: The spectrum of the transfer operator; 2.1 Transfer operators for smooth expanding maps on Hölder functions; 2.2 Transfer operators for smooth expanding maps on Sobolev spaces; 2.2.1 Isotropic Sobolev spaces Htp and good systems of charts
 2.2.2 Bounding the essential spectral radius (Theorem 2.15)2.2.3 The key local LasotaYorke bound (Lemma 2.21); 2.2.4 Fragmentation and reconstitution: Technical lemmas; 2.3 The essential spectral radius on Sobolev spaces: Interpolation; 2.3.1 Complex interpolation; 2.3.2 Proof of Theorem 2.15 on Htp for integer differentiability; 2.4 The essential spectral radius: Dyadic decomposition; 2.4.1 A PaleyLittlewood description of Htp and Ct*; 2.4.2 Proof of Lemma 2.21 and Theorem 2.15: The general case; 2.5 Spectral stability and linear response à la GouëzelKellerLiverani; Problems; Comments
 3 Smooth expanding maps: Dynamical determinants3.1 Ruelle's theorem on the dynamical determinant; 3.1.1 Dynamical zeta functions; 3.2 Ruelle's theorem via kneading determinants; 3.2.1 Outline; 3.2.2 Flat traces; 3.3 Dynamical determinants: Completing the proof of Theorem 3.5; 3.3.1 Proof of Theorem 3.5 if >d+t; 3.3.2 Nuclear power decomposition via approximation numbers; 3.3.3 Asymptotic vanishing of flat traces of the noncompact term; 3.3.4 The case d+t of low differentiability; Problems; Comments; Part II Smooth hyperbolic maps; 4 Anisotropic Banach spaces defined via cones
 4.1 Transfer operators for hyperbolic dynamics4.1.1 Hyperbolic dynamics and anisotropic spaces; 4.1.2 Bounding the essential spectral radius (Theorem 4.6); 4.1.3 Reducing to the transitive case; 4.2 The spaces Wp,*t,s and Wp,**t,s; 4.2.1 Charts and cone systems adapted to (T,V); 4.2.2 Formal definition of the spaces Wp,*t,s and Wp,**t,s; 4.3 The local LasotaYorke lemma and the proof of Theorem 4.6; 4.3.1 The PaleyLittlewood description of the spaces and the local LasotaYorke lemma; 4.3.2 Fragmentation, reconstitution, and the proof of Theorem 4.6; Problems; Comments
 5 A variational formula for the essential spectral radius5.1 Yet another anisotropic Banach space: Bt,s; 5.1.1 Defining Bt,s; 5.2 Bounding the essential spectral radius on Bt,s (Theorem 5.1); 5.3 Spectral stability and linear response; Problems; Comments; 6 Dynamical determinants for smooth hyperbolic dynamics; 6.1 Dynamical determinants via regularised determinants and flat traces; 6.2 Proof of Theorem 6.2 on dT,g(z) if r1> d+ ts; 6.3 Theorem 6.2 in low differentiability r1d+ts; 6.4 Operators on vector bundles and dynamical zeta functions; Problems; Comments
 Extent
 1 online resource (xv, 291 pages)
 Form of item
 online
 Isbn
 9783319776613
 Media category
 computer
 Media MARC source
 rdamedia
 Media type code

 c
 Other control number
 10.1007/9783319776613
 Other physical details
 illustration.
 System control number

 on1035635786
 (OCoLC)1035635786
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