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" Dynamical Zeta functions and dynamical determinants for hyperbolic maps : "
Viviane Baladi.
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BL
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| Record Number
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865468
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| Main Entry
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Baladi, Viviane
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| Title & Author
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Dynamical Zeta functions and dynamical determinants for hyperbolic maps : : a functional approach /\ Viviane Baladi.
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| Publication Statement
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Cham, Switzerland :: Springer,, 2018.
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| Series Statement
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Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics,; volume 68
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1 online resource (xv, 291 pages) :: illustration
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| ISBN
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3319776614
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: 9783319776613
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3319776606
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9783319776606
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| Bibliographies/Indexes
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Includes bibliographical references and index.
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| Contents
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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.
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2.2.2 Bounding the essential spectral radius (Theorem 2.15)2.2.3 The key local Lasota-Yorke 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 Paley-Littlewood 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 Gouëzel-Keller-Liverani; Problems; Comments.
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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 non-compact 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.
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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 Lasota-Yorke lemma and the proof of Theorem 4.6; 4.3.1 The Paley-Littlewood description of the spaces and the local Lasota-Yorke lemma; 4.3.2 Fragmentation, reconstitution, and the proof of Theorem 4.6; Problems; Comments.
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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 r-1> d+ t-s; 6.3 Theorem 6.2 in low differentiability r-1d+t-s; 6.4 Operators on vector bundles and dynamical zeta functions; Problems; Comments.
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| Abstract
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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.
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| Subject
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Banach spaces.
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Dynamics.
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Functions, Zeta.
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Geometry, Hyperbolic.
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Banach spaces.
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Dynamics.
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Functional analysis transforms.
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Functions, Zeta.
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Geometry, Hyperbolic.
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MATHEMATICS-- Calculus.
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MATHEMATICS-- Mathematical Analysis.
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Nonlinear science.
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| Dewey Classification
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515/.56
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| LC Classification
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QA351
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