A First Course in Ergodic Theory

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