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El. knyga: Ergodic Theory and Dynamical Systems

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  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Nov-2016
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781447172871
Kitos knygos pagal šią temą:
Ergodic Theory and Dynamical SystemsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Universitext
  • Išleidimo metai: 10-Nov-2016
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781447172871
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This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics.

This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors.

Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader.

Recenzijos

This textbook is addressed to graduate students as well as to researchers who are not experts in ergodic theory and theory of dynamical systems. For an introduction to the subject it is a very good modern source. (Ivan Podvigin, zbMATH, August, 2017) 

This 200-page book covers most relevant topics for a course in ergodic theory and dynamical systems, addressing topological and measure theoretic perspectives, and including notions of entropy. The subjects are illustrated with selected examples and bibliographical notes on the development of the theory. It is a delightful brief introduction, designed to be a course book on the theme. (Tślio O. Carvalho, Mathematical Reviews, August, 2017)

Part I Ergodic Theory
1 The Mean Ergodic Theorem
3(12)
1.1 Introduction
3(1)
1.2 The Mean Ergodic Theorem
4(4)
1.3 Application to Classical Mechanics
8(4)
1.4 Exercises
12(1)
1.4.1 Basic Exercises
12(1)
1.4.2 More Advanced Exercises
13(1)
1.5 Comments
13(2)
2 The Pointwise Ergodic Theorem
15(10)
2.1 Introduction
15(1)
2.2 The Pointwise Ergodic Theorem
16(4)
2.3 Ergodicity of the Shift
20(3)
2.4 Exercises
23(1)
2.4.1 Basic Exercises
23(1)
2.4.2 More Advanced Exercises
23(1)
2.5 Comments
24(1)
3 Mixing
25(10)
3.1 Introduction
25(1)
3.2 Definition of Mixing
26(1)
3.3 Example: Multiplication by 2
27(1)
3.4 Example: The Bernoulli Shift
28(1)
3.5 Example: Toral Endomorphisms
28(3)
3.6 Exercises
31(1)
3.6.1 Basic Exercises
31(1)
3.6.2 More Advanced Exercises
31(1)
3.7 Comments
32(3)
4 The Hopf Argument
35(14)
4.1 Introduction
35(1)
4.2 Stable Foliation and Invariant Functions
36(2)
4.3 Application to Toral Automorphisms
38(1)
4.4 Flows on the Quotients of PSL2 (R)
39(4)
4.5 Exercises
43(1)
4.5.1 Basic Exercises
43(1)
4.5.2 More Advanced Exercises
44(1)
4.6 Comments
44(5)
Part II Dynamical Systems
5 Topological Dynamics
49(10)
5.1 Introduction
49(1)
5.2 Transitivity and Topological Mixing
50(2)
5.3 Recurrent Points and the Nonwandering Set
52(3)
5.4 Exercises
55(1)
5.4.1 Basic Exercises
55(1)
5.4.2 More Advanced Exercises
55(1)
5.5 Comments
56(3)
6 Nonwandering
59(10)
6.1 Introduction
59(1)
6.2 Nonwandering
60(1)
6.3 Examples
61(2)
6.4 The Graph Associated with the Dynamical System
63(2)
6.5 Exercises
65(1)
6.5.1 Basic Exercises
65(1)
6.5.2 More Advanced Exercises
65(1)
6.6 Comments
66(3)
7 Conjugation
69(10)
7.1 Introduction
69(1)
7.2 Conjugation and Semiconjugation
70(2)
7.3 Elliptic Functions
72(1)
7.4 The Simple Pendulum
73(1)
7.5 Schroder's Examples (1871)
74(2)
7.6 Exercises
76(1)
7.6.1 Basic Exercises
76(1)
7.6.2 More Advanced Exercises
76(1)
7.7 Comments
77(2)
8 Linearization
79(10)
8.1 Introduction
79(1)
8.2 The Hyperbolic Fixed Point Theorem
80(1)
8.3 The Linearization Theorem, Lipschitz Case
80(2)
8.4 The Linearization Theorem, Differentiable Case
82(4)
8.5 Exercises
86(1)
8.5.1 Basic Exercises
86(1)
8.5.2 More Advanced Exercises
86(1)
8.6 Comments
87(2)
9 A Strange Attractor
89(12)
9.1 Introduction
89(1)
9.2 Perturbation of a Toral Automorphism
90(2)
9.3 Perturbed Dynamics
92(1)
9.4 Transitivity and the Mixing Property
93(3)
9.5 Exercises
96(1)
9.5.1 Basic Exercises
96(1)
9.5.2 More Advanced Exercises
96(1)
9.6 Comments
97(4)
Part III Entropy Theory
10 Entropy
101(12)
10.1 Introduction
101(1)
10.2 Definition of Entropy
102(2)
10.3 Properties of Entropy
104(1)
10.4 Generating Partitions
105(2)
10.5 Entropy and Isomorphisms
107(3)
10.6 Exercises
110(1)
10.6.1 Basic Exercises
110(1)
10.6.2 More Advanced Exercises
111(1)
10.7 Comments
111(2)
11 Entropy and Information Theory
113(10)
11.1 Introduction
113(1)
11.2 The Notion of Information
114(1)
11.3 The Game of Questions and Answers
115(1)
11.4 Information and Markov Chains
115(3)
11.5 Interpretation in the Dynamical Setting
118(1)
11.6 Exercises
119(1)
11.6.1 Basic Exercises
119(1)
11.6.2 More Advanced Exercise
120(1)
11.7 Comments
120(3)
12 Computing Entropy
123(12)
12.1 Introduction
123(1)
12.2 The Rokhlin Formula
124(2)
12.3 Entropy of Shifts
126(1)
12.4 Entropy of Dilating Transformations
127(3)
12.5 Exercises
130(1)
12.5.1 Basic Exercises
130(1)
12.5.2 More Advanced Exercises
131(1)
12.6 Comments
131(4)
Part IV Ergodic Decomposition
13 Lebesgue Spaces and Isomorphisms
135(10)
13.1 Introduction
135(1)
13.2 Measurable Isomorphism
136(2)
13.3 Lebesgue Spaces
138(2)
13.4 The Measurable Stone--Weierstra B Theorem
140(2)
13.5 Exercises
142(1)
13.5.1 Basic Exercises
142(1)
13.5.2 More Advanced Exercises
142(1)
13.6 Comments
143(2)
14 Ergodic Decomposition
145(10)
14.1 Introduction
145(1)
14.2 Disintegration
146(2)
14.3 Ergodic Decomposition
148(4)
14.4 Exercises
152(1)
14.4.1 Basic Exercises
152(1)
14.4.2 More Advanced Exercises
153(1)
14.5 Comments
153(2)
15 Measurable Partitions and σ-Algebras
155(12)
15.1 Introduction
155(1)
15.2 Measurable Partitions
156(1)
15.3 The σ-Algebra Associated with a Partition
157(1)
15.4 The Partition Associated with a σ-Algebra
157(2)
15.5 Factors and Partitions
159(1)
15.6 σ-Algebras and Algebras of Functions
160(1)
15.7 The Rokhlin Correspondence
160(2)
15.8 Exercises
162(5)
15.8.1 Basic Exercises
162(1)
15.8.2 More Advanced Exercises
163(4)
Part V Appendices
16 Weak Convergence
167(4)
16.1 Convergence in a Hilbert Space
167(1)
16.2 Weak Sequential Compactness
168(1)
16.3 Convex Closed Subsets
169(2)
17 Conditional Expectation
171(4)
17.1 Definition of the Conditional Expectation
171(1)
17.2 Properties of the Conditional Expectation
172(1)
17.3 The Martingale Convergence Theorem in L2
172(3)
18 Topology and Measures
175(6)
18.1 Separability
175(1)
18.2 The Support of a Measure
175(1)
18.3 Density in the LP Spaces
176(2)
18.4 Inner Regularity
178(1)
18.5 Exercises
179(2)
Notation 181(2)
References 183(2)
Index 185(4)
Author Index 189
Yves Coudčne is a full professor at Brest University, France. His research areas include hyperbolic dynamics, ergodic theory and the geometry of negatively curved spaces.
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