Dynamical Systems and Ergodic Theory

by Mark Pollicott , Michiko Yuri

• ISBN13: 9780521572941
• ISBN10: 0521572940
• Format: Hardcover
• Copyright: 1998-02-13
• Publisher: Cambridge University Press

Summary

This book is essentially a self-contained introduction to topological dynamics and ergodic theory. It is divided into a number of relatively short chapters with the intention that each may be used as a component of a lecture course tailored to the particular audience. Parts of the book are suitable for a final year undergraduate course or for a master's level course. A number of applications are given, principally to number theory and arithmetic progressions (through van der Waerden's theorem and Szemerdi's theorem).

Table of Contents

Introduction	ix	(2)
Preliminaries	xi
1. Conventions	xi	(1)
2. Notation	xi	(1)
3. Pre-requisities in point set topology (Chapters 1-6)	xi	(1)
4. Pre-requisities in measure theory (Chapters 7-12)	xii	(1)
5. Subadditive sequences	xiii	(1)
6. References	xiii
Chapter 1. Examples and basic properties	1	(10)
1.1. Examples	1	(1)
1.2. Transitivity	2	(2)
1.3. Other characterizations of transitivity	4	(1)
1.4. Transitivity for subshifts of finite type	5	(1)
1.5. Minimality and the Birkhoff recurrence theorem	6	(2)
1.6. Commuting homeomorphisms	8	(1)
1.7. Comments and references	9	(2)
Chapter 2. An application of recurrence to arithmetic progressions	11	(8)
2.1. Van der Waerden's theorem	11	(1)
2.2. A dynamical proof	12	(3)
2.3. The proofs of Sulemma 2.2.2 and Sublemma 2.2.3	15	(2)
2.4. Comments and references	17	(2)
Chapter 3. Topological entropy	19	(14)
3.1. Definitions	19	(4)
3.2. The Perron-Frobenius theorem and subshifts of finite type	23	(3)
3.3. Other definitions and examples	26	(4)
3.4. Conjugacy	30	(2)
3.5. Comments and references	32	(1)
Chapter 4. Interval maps	33	(14)
4.1. Fixed points and periodic points	33	(4)
4.2. Topological entropy of interval maps	37	(2)
4.3. Markov maps	39	(5)
4.4. Comments and references	44	(3)
Chapter 5. Hyperbolic toral automorphisms	47	(10)
5.1. Definitions	47	(2)
5.2. Entropy for Hyperbolic Toral Automorphisms	49	(3)
5.3. Shadowing and semi-conjugacy	52	(3)
5.4. Comments and references	55	(2)
Chapter 6. Rotation numbers	57	(8)
6.1. Homeomorphisms of the circle and rotation numbers	57	(3)
6.2. Denjoy's theorem	60	(4)
6.3. Comments and references	64	(1)
Chapter 7. Invariant measures	65	(8)
7.1. Definitions and characterization of invariant measures	65	(1)
7.2. Borel sigma-algebras for compact metric spaces	65	(2)
7.3. Examples of invariant measures	67	(2)
7.4. Invariant measures for other actions	69	(2)
7.5. Comments and references	71	(2)
Chapter 8. Measure theoretic entropy	73	(18)
8.1. Partitions and conditional expectations	73	(3)
8.2. The entropy of a partition	76	(3)
8.3. The entropy of a transformation	79	(3)
8.4. The increasing martingale theorem	82	(2)
8.5. Entropy and sigma algebras	84	(2)
8.6. Conditional entropy	86	(1)
8.7. Proofs of Lemma 8.7 and Lemma 8.8	87	(1)
8.8. Isomorphism	88	(1)
8.9. Comments and references	89	(2)
Chapter 9. Ergodic measures	91	(8)
9.1. Definitions and characterization of ergodic measures	91	(1)
9.2. Poincare recurrence and Kac's theorem	91	(2)
9.3. Existence of ergodic measures	93	(1)
9.4. Some basic constructions in ergodic theory	94	(3)
9.4.1. Skew products	95	(1)
9.4.2. Induced transformations and Rohlin towers	95	(1)
9.4.3. Natural extensions	96	(1)
9.5. Comments and references	97	(2)
Chapter 10. Ergodic theorems	99	(14)
10.1. The Von Neumann ergodic theorem	99	(3)
10.2. The Birkhoff theorem (for ergodic measures)	102	(4)
10.3. Applications of the ergodic theorems	106	(5)
10.4. The Birkhoff theorem (for invariant measures)	111	(1)
10.5. Comments and references	112	(1)
Chapter 11. Mixing Properties	113	(12)
11.1. Weak mixing	113	(1)
11.2. A density one convergence characterization of weak mixing	114	(2)
11.3. A generalization of the Von Neumann ergodic theorem	116	(2)
11.4. The spectral viewpoint	118	(2)
11.5. Spectral characterization of weak mixing	120	(2)
11.6. Strong mixing	122	(1)
11.7. Comments and reference	123	(2)
Chapter 12. Statistical properties in ergodic theory	125	(14)
12.1. Exact endomorphisms	125	(1)
12.2. Statistical properties of piecewise expanding Markov maps	126	(7)
12.3. Rohlin's entropy formula	133	(1)
12.4. The Shannon-McMillan-Brieman theorem	134	(3)
12.5. Comments and references	137	(2)
Chapter 13. Fixed points for homeomorphisms of the annulus	139	(8)
13.1. Fixed points for the annulus	139	(5)
13.2. Outline proof of Brouwer's theorem	144	(2)
13.3. Comments and references	146	(1)
Chapter 14. The variational principle	147	(6)
14.1. The variational principle for entropy	147	(1)
14.2. The proof of the variational principle	147	(5)
14.3. Comments and reference	152	(1)
Chapter 15. Invariant measures for commuting transformations	153	(8)
15.1. Furstenberg's conjecture and Rudolph's theorem	153	(1)
15.2. The proof of Rudolph's theorem	153	(6)
15.3. Comments and references	159	(2)
Chapter 16. Multiple recurrence and Szemeredi's theorem	161	(16)
16.1. Szemeredi's theorem on arithmetic progressions	161	(1)
16.2. An ergodic proof of Szemeredi's theorem	162	(1)
16.3. The proof of Theorem 16.2	163	(3)
16.3.1. (UMR) for weak-mixing systems, weak-mixing extensions and compact systems	163	(2)
16.3.2. The non-weak-mixing case	165	(1)
16.3.3. (UMR) for compact extensions	165	(1)
16.3.4. The last step	165	(1)
16.4. Appendix to section 16.3	166	(10)
16.4.1. The proofs of Propositions 16.3 and 16.4	166	(5)
16.4.2. The proof of Proposition 16.5	171	(1)
16.4.3. The proof of Proposition 16.6	171	(1)
16.4.4. The proof of Proposition 16.7	172	(1)
16.4.5. The proof of Proposition 16.8	173	(2)
16.4.6. The proof of Proposition 16.9	175	(1)
16.5. Comments and references	176	(1)
Index	177