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Entropy in Dynamical Systems.
A comprehensive course on entropy in dynamical systems ideal for graduate students learning the subject from scratch.
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cambridge :
Cambridge University Press,
2011.
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| Colección: | New mathematical monographs.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Cover; Entropy in Dynamical Systems; NEW MATHEMATICAL MONOGRAPHS; Title; Copyright; To my Parents; Contents; Preface; Acknowledgments; Introduction; 0.1 The leitmotiv; 0.2 A few words about the history of entropy; 0.3 Multiple meanings of entropy; 0.3.1 Entropy in physics; 0.3.2 Shannon entropy; 0.3.3 Connection between Shannon and Boltzmann entropy; 0.3.4 Dynamical entropy; 0.3.5 Dynamical entropy as data compression rate; 0.3.6 Entropy as disorder; 0.4 Conventions; Part I Entropy in ergodic theory; 1 Shannon information and entropy; 1.1 Information and entropy of probability vectors.
- 1.2 Partitions and sigma-algebras1.3 Information and static entropy of a partition; 1.4 Conditional static entropy; 1.5 Conditional entropy via probabilistic tools*; 1.6 Basic properties of static entropy; 1.7 Metrics on the space of partitions; 1.8 Mutual information*; 1.9 Non-Shannon inequalities*; Exercises; 2 Dynamical entropy of a process; 2.1 Subadditivity; 2.2 Preliminaries on dynamical systems; 2.3 Dynamical entropy of a process; 2.4 Properties of dynamical entropy; 2.5 Affinity of dynamical entropy; 2.6 Conditional dynamical entropy via disintegration*
- 2.7 Summary of the properties of entropy2.8 Combinatorial entropy; Exercises; 3 Entropy theorems in processes; 3.1 Independence and e-independence; 3.2 The Pinsker sigma-algebra in a process; 3.3 The Shannon-McMillan-Breiman Theorem; 3.4 The Ornstein-Weiss Return Times Theorem; 3.5 Horizontal data compression; Exercises; 4 Kolmogorov-Sinai Entropy; 4.1 Entropy of a dynamical system; 4.2 Generators; 4.3 The natural extension; 4.4 Joinings; 4.5 Ornstein Theory*; Exercises; 5 The Ergodic Law of Series*; 5.1 History of the Law of Series; 5.2 Attracting and repelling in signal processes.
- 5.3 Decay of repelling in positive entropy5.3.1 The idea of the proof and the basic lemma; 5.3.2 The proof of Theorem 5.3.3; 5.4 Typicality of attracting for long cylinders; Part II Entropy in topological dynamics; 6 Topological entropy; 6.1 Three definitions of topological entropy; 6.1.1 The metric definition via separated orbits; 6.1.2 The metric definition via spanning orbits; 6.1.3 The topological definition via covers; 6.1.4 Relations between the above notions; 6.2 Properties of topological entropy; 6.3 Topological conditional and tail entropies.
- 6.4 Properties of topological conditional entropy6.5 Topological joinings; 6.6 The simplex of invariant measures; 6.7 Topological fiber entropy; 6.8 The major Variational Principles; 6.9 Determinism in topological systems; 6.9.1 Topological analogs of determinism; 6.9.2 Hierarchy of maximal factors; 6.10 Topological preimage entropy*; Exercises; 7 Dynamics in dimension zero; 7.1 Zero-dimensional dynamical systems; 7.2 Topological entropy in dimension zero; 7.3 The invariant measures in dimension zero; 7.4 The Variational Principle in dimension zero.