Author(s): Karma Dajani, Charlene Kalle
Publisher: Chapman & Hall / CRC
Year: 2021
Language: English
City: Boca Raton
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
