A First Course in Ergodic Theory

Description

A First Course in Ergodic Theory provides readers with an introductory course in Ergodic Theory. This textbook has been developed from the authors' own notes on the subject, which they have been teaching since the 1990s. Over the years they have added topics, theorems, examples and explanations from various sources. The result is a book that is easy to teach from and easy to learn from - designed to require only minimal prerequisites. Features Suitable for readers with only a basic knowledge of measure theory, some topology and a very basic knowledge of functional analysis Perfect as the primary textbook for a course in Ergodic Theory Examples are described and are studied in detail when new properties are presented. Cover Half Title Title Page Copyright Page Dedication Contents Preface Author Bios Chapter 1: Measure Preservingness and Basic Examples 1.1. WHAT IS ERGODIC THEORY? 1.2. MEASURE PRESERVING TRANSFORMATIONS 1.3. BASIC EXAMPLES Chapter 2: Recurrence and Ergodi 2.1. RECURRENCE 2.2. ERGODICITY 2.3. EXAMPLES OF ERGODIC TRANSFORMATIONS Chapter 3: The Pointwise Ergodic Theorem and Mixing 3.1. THE POINTWISE ERGODIC THEOREM 3.2. NORMAL NUMBERS 3.3. IRREDUCIBLE MARKOV CHAINS 3.4. MIXING Chapter 4: More Ergodic Theorems 4.1. THE MEAN ERGODIC THEOREM 4.2. THE HUREWICZ ERGODIC THEOREM Chapter 5: Isomorphisms and Factor Maps 5.1. MEASURE PRESERVING ISOMORPHISMS 5.2. FACTOR MAPS 5.3. NATURAL EXTENSIONS Chapter 6: The Perron-Frobenius Operator 6.1. ABSOLUTELY CONTINUOUS INVARIANT MEASURES 6.2. EXACTNESS 6.3. PIECEWISE MONOTONE INTERVAL MAPS Chapter 7: Invariant Measures for Continuous Transformations 7.1. EXISTENCE 7.2. UNIQUE ERGODICITY AND UNIFORM DISTRIBUTION 7.3. SOME TOPOLOGICAL DYNAMICS Chapter 8: Continued Fractions 8.1. REGULAR CONTINUED FRACTIONS 8.2. ERGODIC PROPERTIES OF THE GAUSS MAP 8.3. THE DOEBLIN-LENSTRA CONJECTURE 8.4. OTHER CONTINUED FRACTION TRANSFORMATIONS Chapter 9: Entropy 9.1. RANDOMNESS AND INFORMATION 9.2. DEFINITIONS AND PROPERTIES 9.3. CALCULATION OF ENTROPY AND EXAMPLES 9.4. THE SHANNON-MCMILLAN-BREIMAN THEOREM 9.5. LOCHS’ THEOREM Chapter 10: The Variational Principle 10.1. TOPOLOGICAL ENTROPY 10.2. PROOF OF THE VARIATIONAL PRINCIPLE 10.3. MEASURES OF MAXIMAL ENTROPY Chapter 11: Infinite Ergodic Theory 11.1. EXAMPLES 11.2. CONSERVATIVE AND DISSIPATIVE PART 11.3. INDUCED SYSTEMS 11.4. JUMP TRANSFORMATIONS 11.5. INFINITE ERGODIC THEOREMS Chapter 12: Appendix 12.1. TOPOLOGY 12.2. MEASURE THEORY 12.3. LEBESGUE SPACES 12.4. LEBESGUE INTEGRATION 12.5. HILBERT SPACES 12.6. BOREL MEASURES ON COMPACT METRIC SPACES 12.7. FUNCTIONS OF BOUNDED VARIATION Bibliography Index