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Topics in Ergodic Theory
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Iakov Grigorevich Sinai
Language:
English
Published/Copyright:
2017
About this book
This book concerns areas of ergodic theory that are now being intensively developed. The topics include entropy theory (with emphasis on dynamical systems with multi-dimensional time), elements of the renormalization group method in the theory of dynamical systems, splitting of separatrices, and some problems related to the theory of hyperbolic dynamical systems.
Originally published in 1993.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Author / Editor information
Ya. G. Sinai is Professor of Mathematics at Princeton University.
Reviews
"No other book that I can think of covers this range of topics with the level of clarity that this book does.... I can't imagine a better choice than this book for someone wanting to come up to speed with modern developments in mathematical ergodic theory."
Topics
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Frontmatter
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Contents
v -
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Preface
vii - Part I. General Ergodic Theory
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Lecture 1. Measurable Transformations. Invariant Measures. Ergodic Theorems
1 -
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Lecture 2. Lebesgue Spaces and Measurable Partitions. Ergodicity and Decomposition into Ergodic Components. Spectrum of Interval Exchange Transformations
16 -
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Lecture 3. Isomorphism of Dynamical Systems. Generators of Dynamical Systems
28 -
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Lecture 4. Dynamical Systems with Pure Point Spectra
36 -
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Lecture 5. General Properties of Eigenfunctions and Eigenvalues of Ergodic Automorphisms. Isomorphism of Dynamical Systems with Pure Point Spectrum
43 - Part II. Entropy Theory of Dynamical Systems
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Lecture 6. Entropy Theory of Dynamical Systems
53 -
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Lecture 7. Breiman Theorem. Pinsker Partition. K-Systems. Exact Endomorphisms. Gibbs Measures
69 -
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Lecture 8. Entropy of Dynamical Systems with Multidimensional Time. Systems of Cellular Automata as Dynamical Systems
77 - Part III. One-Dimensional Dynamics
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Lecture 9. Continued Fractions and Farey Fractions
85 -
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Lecture 10. Homeomorphisms and Diffeomorphisms of the Circle
95 -
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Lecture 11. Sharkovski's Ordering and Feigenbaum's Universality
111 -
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Lecture 12. Expanding Mappings of the Circle
123 - Part IV. Two-Dimensional Dynamics
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Lecture 13. Standard Map. Twist Maps. Periodic Orbits. Aubry-Mather Theory
135 -
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Lecture 14. Periodic Hyperbolic Points, Their Stable and Unstable Manifolds. Homoclinic and Heteroclinic Orbits
147 -
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Lecture 15. Homoclinic and Heteroclinic Points and Stochastic Layers
167 - Part V. Elements of the Theory of Hyperbolic Dynamical Systems
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Lecture 16. Geodesic Flows and Their Generalizations. Discontinuous Dynamical Systems. Stable and Unstable Manifolds
175 -
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Lecture 17. Existence of Local Manifolds. Gibbs Measures
194 -
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Lecture 18. Markov Partitions. H-Theorem for Dynamical Systems. Elements of Thermodynamic Formalism
204 -
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Index
217
Publishing information
Pages and Images/Illustrations in book
eBook published on:
September 25, 2017
eBook ISBN:
9781400887255
Pages and Images/Illustrations in book
eBook ISBN:
9781400887255
Keywords for this book
Theorem; Ergodic theory; Automorphism; Dynamical system; Existential quantification; Lebesgue measure; Endomorphism; Measure (mathematics); Dimension; Eigenfunction; Mathematical induction; Invariant measure; Recurrence relation; Fubini's theorem; Fundamental lemma (Langlands program); Bifurcation theory; Probability measure; Eigenvalues and eigenvectors; Central limit theorem; Hilbert space; Phase space; Cauchy's integral formula; Poincaré recurrence theorem; Borel–Cantelli lemma; Hamiltonian mechanics; Liouville's theorem (Hamiltonian); Cyclic group; Feigenbaum constants; Schrödinger equation; Continuous function (set theory); Geodesic; Rotation number; Probability distribution; Submanifold; Frenet–Serret formulas; Lebesgue space; Periodic point; Perturbation theory (quantum mechanics); Characterization (mathematics); Subgroup; Functional equation; Riemannian manifold; Symplectic geometry; Semigroup; Permutation; Indicator function; Renormalization group; Variable (mathematics); Equation; Infimum and supremum; Ergodicity; Haar measure; Subsequence; Dimension (vector space); Symbolic dynamics; Subset; Periodic function; Fundamental class; Stochastic; Lorenz system; Theory; Vector bundle; Transitive relation; Analytic continuation; Character group; Intrinsic metric; Linearization; Invariant subspace; Hyperbolic point; Piecewise
Audience(s) for this book
For an expert adult audience, including professional development and academic research
