CONTENTS
PREFACE
Chapter 1: The Orbits of One-Dimensional Maps
1.1: Iteration of Functions and Examples of Dynamical Systems
Exercises
1.2: Newton’s Method and Fixed Points
Exercises
1.3: Graphical Iteration
1.4: The Stability of Fixed Points
Exercises
1.5: Non-Hyperbolic Fixed Points
Exercises
Chapter 2: Bifurcations and the Logistic Family
2.1: The Basin of Attraction
2.2: The Logistic Family
Exercises
2.3: Periodic Points
Exercises
2.4: Periodic Points of the Logistic Map
Exercises
2.5: The Period Doubling Route to Chaos
2.5.1: A Super-Attracting 2-Cycle
2.6: The Bifurcation Diagram and 3-Cycles of the Logistic Map
2.6.1: Where Does Period 3 Occur for the Logistic Map?
2.6.2: A Super-Attracting 3-Cycle for the Logistic Map
2.6.3: The 3-Cycle when μ = 4
Exercises
2.7: The Tent Family T μ
2.8: The 2-Cycles and 3-Cycles of the Tent Family
Exercises
Chapter 3: Sharkovsky’s Theorem
3.1: Period 3 Implies Chaos
3.2: Converse of Sharkovsky’s Theorem
Exercises
Chapter 4: Dynamics on Metric Spaces
4.1: Basic Properties of Metric Spaces
4.2: Dense Sets
Exercises
4.3: Functions between Metric Spaces
Exercises
4.4: Diffeomorphisms of R
Exercises
Chapter 5: Countability, Sets of Measure Zero and the Cantor Set
5.1: Countability and Sets of Measure Zero
5.2: The Cantor Set
Exercises
5.3: Ternary Expansions and the Cantor Set
Exercises
5.4: The Tent Map for μ = 3
5.5: A Cantor Set Arising from the Logistic Map L μ, μ > 4
Chapter 6: Devaney’s Definition of Chaos
6.1: The Doubling Map and the Angle Doubling Map
6.2: Transitivity
6.3: Sensitive Dependence on Initial Conditions
6.4: The Definition of Chaos
6.5: Symbolic Dynamics and the Shift Map
Exercises
6.6: For Continuous Maps, Sensitive Dependence is Implied by Transitivity and Dense Period Points
Exercises
Chapter 7: Conjugacy of Dynamical Systems
7.1: Conjugate Maps
Exercises
7.2: Properties of Conjugate Maps and Chaos Through Conjugacy
Exercises
7.3: Linear Conjugacy
Exercises
Chapter 8: Singer’s Theorem
8.1: The Schwarzian Derivative Revisited
8.2: Singer’s Theorem
Exercises
Chapter 9: Conjugacy, Fundamental Domains and the Tent Family
9.1: Conjugacy and Fundamental Domains
Exercises
9.2: Conjugacy, the Tent Map and Periodic Points of the Tent Family
Exercises
Chapter 10: Fractals
10.1: Examples of Fractals
10.2: An Intuitive Introduction to the Idea of Fractal Dimension
10.3: Box Counting Dimension
Exercises
10.4: The Mathematical Theory of Fractals
10.4.1: Complete Metric Spaces
Exercises
10.5: The Contraction Mapping Theorem and Self-Similar Sets
Exercises
Chapter 11: Newton’s Method for Real Quadratics and Cubics
11.1: Binary Representation of Real Numbers
11.2: Newton’s Method for Real Quadratic Polynomials
11.3: Newton’s Method for Real Cubic Polynomials
11.4: The Cubic Polynomials f c(x) = (x+2)(x2 + c)
Exercises
Chapter 12: Coppel’s Theorem and a Proof of Sharkovsky’s Theorem
12.1: Coppel’s Theorem
12.2: The Proof of Sharkovsky’s Theorem
12.3: The Completion of the Proof of Sharkovsky’s Theorem
12.3.1: Parts (a), (b) and (c) in Theorem 12.2.2 Imply Sharkovsky’s Theorem
12.3.2: Proof of Sharkovsky’s Theorem Parts (ii) and (iii)
Exercises
Chapter 13: Real Linear Transformations, the Hénon Map and Hyperbolic Toral Automorphisms
13.1: Linear Transformations
Exercises
13.2: The Hénon Map
13.2.1: The Hénon Attractor
Exercises
13.3: Circle Maps Induced by Linear Transformations on R
13.4: Endomorphisms of the Torus
Exercises
13.5: Hyperbolic Toral Automorphisms
Exercises
Chapter 14: Elementary Complex Dynamics
14.1: The Complex Numbers
14.2: Analytic Functions in the Complex Plane
Exercises
14.3: The Dynamics of Polynomials and the Riemann Sphere
Exercises
14.4: The Julia Set
14.4.1: Properties of the Julia Set of a Polynomial
14.4.2: The Quadratic Maps fc(z) = z2 + c
14.4.3: Maps for Which the Sequence (F nc (0)) is Unbounded
Exercises
14.5: The Mandelbrot Set M
14.5.1: Properties of the Mandelbrot Set
14.5.2: Computer Graphics for the Mandelbrot Set
Exercises
14.6: Newton’s Method in the Complex Plane for Quadratics and Cubics
14.6.1: Cubic Polynomials
14.6.2: The Basin of Attraction of the Newton Function of a Polynomial Having At Least Three Distinct Roots
14.6.3: The cubic pα(z)=z(z-1)(z-α)
Exercises
14.7: Important Complex Functions
14.7.1: The Exponential Function: ez
14.7.2: The Complex Logarithm Function: Log (z)
14.7.3: The Complex Arctangent Function: Arctan(z)
14.7.4: Halley’s Method and the Schröder–Cayley Theorem
Exercises
Chapter 15: Examples of Substitutions
15.1: One-Dimensional Substitutions and the Thue–Morse Substitution
15.1.1: The Thue–Morse sequence
15.1.2: Properties of the Thue–Morse Sequence u
15.1.3: Definition of a Substitution
Exercises
15.2: The Toeplitz Substitution
Exercises
15.3: The Rudin–Shapiro Sequence
15.3.1: The Rudin–Shapiro Substitution
Exercises
15.4: Paperfolding Sequences
15.4.1: The Substitution Associated with a Paperfolding Sequence
Exercises
Chapter 16: Fractals Arising from Substitutions
16.1: A Connection between the Morse Substitution and the Koch Curve
Exercises
16.2: Dragon Curves
16.3: Fractals Arising from Two-Dimensional Substitutions
16.3.1: Substitution Tilings
Exercises
16.4: The Rauzy Fractal
16.4.1: The Incidence Matrix of a Substitution
16.4.2: The Fibonacci Quasicrystal
16.4.3: Construction of the Rauzy Fractal
16.4.4: Additional Properties for the Rauzy Fractal
Exercises
Chapter 17: Compactness in Metric Spaces and an Introduction to Topological Dynamics
17.1: Compactness in Metric Spaces
Exercises
17.2: Continuous Functions on Compact Metric Spaces
Exercises
17.3: The Contraction Mapping Theorem for Compact Metric Spaces
17.4: Basic Topological Dynamics
17.4.1: Minimality
Exercises
17.5: Topological Mixing and Exactness
Exercises
Chapter 18: Substitution Dynamical Systems
18.1: Sequence Spaces
Exercises
18.2: Languages
18.2.1: Languages and Words
18.2.2: The Complexity Function of a Sequence
Exercises
18.3: Dynamical Systems Arising from Sequences
Exercises
18.4: Substitution Dynamics
Exercises
Chapter 19: Sturmian Sequences and Irrational Rotations
19.1: Sturmian Sequences
Exercises
19.2: Sequences Arising from Irrational Rotations
Exercises
19.3: Cutting Sequences
19.3.1: Billiard Balls in a Square
Exercises
19.4: Sequences Arising from Irrational Rotations are Sturmian
19.5: Semi-Topological Conjugacy between ([0, 1), T α) and (O(u) , σ )
Exercises
19.6: The Three Distance Theorem
Exercises
Chapter 20: The Multiple Recurrence Theorem of Furstenberg and Weiss
20.1: van der Waerden’s Theorem
Exercises
APPENDIX A: THEOREMS FROM CALCULUS
APPENDIX B: THE BAIRE CATEGORY THEOREM
APPENDIX C: THE COMPLEX NUMBERS
APPENDIX D: WEYL’S EQUIDISTRIBUTION THEOREM
REFERENCES
INDEX