This textbook provides a broad introduction to the fields of dynamical systems and ergodic theory. Motivated by examples throughout, the author offers readers an approachable entry-point to the dynamics of ergodic systems. Modern and classical applications complement the theory on topics ranging from financial fraud to virus dynamics, offering numerous avenues for further inquiry.
Starting with several simple examples of dynamical systems, the book begins by establishing the basics of measurable dynamical systems, attractors, and the ergodic theorems. From here, chapters are modular and can be selected according to interest. Highlights include the Perron–Frobenius theorem, which is presented with proof and applications that include Google PageRank. An in-depth exploration of invariant measures includes ratio sets and type III measurable dynamical systems using the von Neumann factor classification. Topological and measure theoretic entropy are illustrated and compared in detail, with an algorithmic application of entropy used to study the papillomavirus genome. A chapter on complex dynamics introduces Julia sets and proves their ergodicity for certain maps. Cellular automata are explored as a series of case studies in one and two dimensions, including Conway’s Game of Life and latent infections of HIV. Other chapters discuss mixing properties, shift spaces, and toral automorphisms.
Ergodic Dynamics unifies topics across ergodic theory, topological dynamics, complex dynamics, and dynamical systems, offering an accessible introduction to the area. Readers across pure and applied mathematics will appreciate the rich illustration of the theory through examples, real-world connections, and vivid color graphics. A solid grounding in measure theory, topology, and complex analysis is assumed; appendices provide a brief review of the essentials from measure theory, functional analysis, and probability.
Author(s): Jane Hawkins
Series: Graduate Texts in Mathematics 289
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 336
City: Cham
Tags: Dynamical Systems, Attractors, Ergodic Theorems, Shift Spaces, Invariant Measures, Entropy, Fractals, Cellular Automata
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
