Ergodic Dynamics: From Basic Theory to Applications

Preface
	Acknowledgments
Contents
1 The Simplest Examples
	1.1 Symbol Spaces and Bernoulli Shifts
	Exercises
2 Dynamical Properties of Measurable Transformations
	2.1 The Basic Definitions
	2.2 Recurrent, Conservative, and Dissipative Systems
		2.2.1 Ergodicity
		2.2.2 Kac's Lemma
		2.2.3 Conservativity and Hopf Decomposition
	2.3 Noninvertible Maps and Exactness
	Exercises
3 Attractors in Dynamical Systems
	3.1 Attractors
	3.2 Examples of Attractors
	3.3 Sensitive Dependence, Chaotic Dynamics, and Turbulence
		3.3.1 Unimodal Interval Maps
	Exercises
4 Ergodic Theorems
	4.1 The Koopman Operator for a Dynamical System
	4.2 Von Neumann Ergodic Theorems
	4.3 Birkhoff Ergodic Theorem
	4.4 Spectrum of an Ergodic Dynamical System
	4.5 Unique Ergodicity
		4.5.1 The Topology of Probability Measures on Compact Metric Spaces
	4.6 Normal Numbers and Benford's Law
		4.6.1 Normal Numbers
		4.6.2 Benford's Law
		4.6.3 Detecting Financial Fraud Using Benford's Law
	Exercises
5 Mixing Properties of Dynamical Systems
	5.1 Weak Mixing and Mixing
	5.2 Noninvertibility
		5.2.1 Partitions
		5.2.2 Rohlin Partitions and Factors
	5.3 The Parry Jacobian and Radon–Nikodym Derivatives
	5.4 Examples of Noninvertible Maps
	5.5 Exact Endomorphisms
	Exercises
6 Shift Spaces
	6.1 Full Shift Spaces and Bernoulli Shifts
	6.2 Markov shifts
		6.2.1 Subshifts of Finite Type
	6.3 Markov Shifts in Higher Dimensions
	6.4 Noninvertible Shifts
		6.4.1 Index Function
	Exercises
7 Perron–Frobenius Theorem and Some Applications
	7.1 Preliminary Background
	7.2 Spectrum and the Perron–Frobenius Theorem
		7.2.1 Application to Markov Shift Dynamics
	7.3 An Application to Google's PageRank
	7.4 An Application to Virus Dynamics
		7.4.1 States of the Markov Process
	Exercises
8 Invariant Measures
	8.1 Measures for Continuous Maps
	8.2 Induced Transformations
	8.3 Existence of Absolutely Continuous Invariant Probability Measures
		8.3.1 Weakly Wandering Sets for Invertible Maps
		8.3.2 Proof of the Hajian–Kakutani Weakly Wandering Theorem
	8.4 Halmos–Hopf–von Neumann Classification
	Exercises
9 No Equivalent Invariant Measures: Type III Maps
	9.1 Ratio Sets
	9.2 Odometers of Type II and Type III
		9.2.1 Krieger Flows
		9.2.2 Type III0 Dynamical Systems
	9.3 Other Examples
		9.3.1 Noninvertible Maps
	Exercises
10 Dynamics of Automorphisms of the Torus and Other Groups
	10.1 An Illustrative Example
	10.2 Dynamical and Ergodic Properties of Toral Automorphisms
	10.3 Group Endomorphisms and Automorphisms on Tn
		10.3.1 Ergodicity and Mixing of Toral Endomorphisms
	10.4 Compact Abelian Group Rotation Dynamics
	Exercises
11 An Introduction to Entropy
	11.1 Topological Entropy
		11.1.1 Defining and Calculating Topological Entropy
		11.1.2 Hyperbolic Toral Endomorphisms
		11.1.3 Topological Entropy of Subshifts
			11.1.3.1 Markov Shifts
	11.2 Measure Theoretic Entropy
		11.2.1 Preliminaries for Measure Theoretic Entropy
		11.2.2 The Definition of hμ(f)
		11.2.3 Computing hμ(f)
			11.2.3.1 Generators
			11.2.3.2 Conditional Entropy
		11.2.4 An Information Theory Derivation of H(P)
	11.3 Variational Principle
	11.4 An Application of Entropy to the Papillomavirus Genome
		11.4.1 Algorithm
	Exercises
12 Complex Dynamics
	12.1 Background and Notation
		12.1.1 Some Dynamical Properties of Iterated Functions
	12.2 Möbius Transformations and Conformal Conjugacy
		12.2.1 The Dynamics of Möbius Transformations
			12.2.1.1 Conformal Conjugacy
	12.3 Julia Sets
		12.3.1 First Properties of J(R)
		12.3.2 Exceptional and Completely Invariant Sets
		12.3.3 Dynamics on Julia Sets
		12.3.4 Classification of the Fatou Cycles
	12.4 Ergodic Properties of Some Rational Maps
		12.4.1 Ergodicity of Non-Critical Postcritically Finite Maps
	Exercises
13 Maximal Entropy Measures on Julia Sets and a Computer Algorithm
	13.1 The Random Inverse Iteration Algorithm
	13.2 Statement of the Results
	13.3 Markov Processes for Rational Maps
		13.3.1 Proof of Theorem 13.2.
	13.4 Proof That the Algorithm Works
	13.5 Ergodic Properties of the Mañé–Lyubich Measure
	13.6 Fine Structure of the Mañé–Lyubich Measure
	Exercises
14 Cellular Automata
	14.1 Definition and Basic Properties
		14.1.1 One-Dimensional CAs
		14.1.2 Notation for Binary CAs with Radius 1
		14.1.3 Topological Dynamical Properties of CA F90
		14.1.4 Measures for CAs
	14.2 Equicontinuity Properties of CA
	14.3 Higher Dimensional CAs
		14.3.1 Conway's Game of Life
	14.4 Stochastic Cellular Automata
	14.5 Applications to Virus Dynamics
	Exercises
A Measures on Topological Spaces
	A.1 Lebesgue Measure on R
		A.1.1 Properties of m
			A.1.1.1 Outer Measure m*
		A.1.2 A Non-measurable Set
	A.2 Sets of Lebesgue Measure Zero
		A.2.1 Examples of Null Sets
		A.2.2 A Historical Note on Lebesgue Measure
	A.3 The Definition of a Measure Space
	A.4 Measures and Topology in Metric Spaces
		A.4.1 Approximation and Extension Properties
			A.4.1.1 Radon Measures on σ-Compact and Locally Compact Metric Spaces
		A.4.2 The Space of Borel Probability Measures on X
		A.4.3 Hausdorff Measures and Dimension
		A.4.4 Some Useful Tools
	A.5 Examples of Metric Spaces with Borel Measures
		A.5.1 One-Dimensional Spaces
		A.5.2 Discrete Measure Spaces
		A.5.3 Product Spaces
		A.5.4 Other Spaces of Interest
			A.5.4.1 Quotient Spaces and Tori
			A.5.4.2 Symbol Spaces
	Exercises
B Integration and Hilbert Spaces
	B.1 Integration
		B.1.1 Conventions About Values at ∞ and Measure 0 Sets
		B.1.2 Lp Spaces
	B.2 Hilbert Spaces
		B.2.1 Orthonormal Sets and Bases
		B.2.2 Orthogonal Projection in a Hilbert Space
	B.3 Von Neumann Factors from Ergodic Dynamical Systems
	Exercises
C Connections to Probability Theory
	C.1 Vocabulary and Notation of Probability Theory
	C.2 The Borel-Cantelli Lemma
	C.3 Weak and Strong Laws of Large Numbers
	Exercises
References
Index