Ergodic Theory (Encyclopedia of Complexity and Systems Science Series)

Series Preface
Volume Preface
Contents
About the Editor-in-Chief
About the Volume Editors
Contributors
Introduction to Ergodic Theory
Ergodic Theory: Basic Examples and Constructions
	Glossary
	Definition of the Subject and its Importance
	Introduction
	Examples
		Rigid Rotation of a Compact Group
		Adding Machines
		Interval Exchange Maps
		Full Shifts and Shifts of Finite Type
		More Examples of Subshifts
			Prouhet-Thue-Morse
			Chacon System
			Sturmian Systems
			Toeplitz Systems
			Sofic Systems
			Context-Free Systems
			Coded Systems
		Smooth Expanding Interval Maps
			Piecewise C2 Expanding Maps
		More Interval Maps
			Continued Fraction Map
			The Farey Map
			F-Expansions
			β-Shifts
		Gaussian Systems
		Hamiltonian Systems
			Billiard Systems
			KAM-Systems and Stably Nonergodic Behavior
		Smooth Uniformly Hyperbolic Diffeomorphisms and Flows
			Geodesic Flow on Manifold of Negative Curvature
			Horocycle Flow
			Markov Partitions and Coding
			Anosov Systems
			Axiom A Systems
				Horseshoe Maps
				Solenoids
		Partially Hyperbolic Dynamical Systems
			Compact Group Extensions of Uniformly Hyperbolic Systems
			Time-One Maps of Anosov Flows
		Nonuniformly Hyperbolic Systems
		Physically Relevant Measures and Strange Attractors
			Unimodal Maps
			Intermittent Maps
			Hénon Diffeomorphisms
		Complex Dynamics
		Infinite Ergodic Theory
	Constructions
		Products
		Factors
		Skew Products
			Random Dynamical Systems
			Group Extensions of Dynamical Systems
		Induced Transformations
		Suspension Flows
		Cutting and Stacking
		Adic Transformations
		Rokhlin´s Lemma
		Inverse Limits
		Natural Extension
		Joinings
	Future Directions
	Bibliography
		Primary Literature
		Books and Reviews
Ergodicity and Mixing Properties
	Glossary
	Introduction
		Basics, Examples, and Highlighted Applications
		Some Iconic Examples
		Factors, Extensions, and Isomorphisms
		Highlighted Applications and Connections
	Ergodicity
		Ergodic Decomposition
	Mixing
	Hyperbolicity and Decay of Correlations
	Representations, Realizations, and Genericity
		Topological and Set-Theoretic Properties of Classes of Transformations
	Future Directions
	References
Ergodic Theory: Recurrence
	Glossary
	Definition of the Subject and Its Importance
	Introduction
	Quantitative Poincaré Recurrence
		Early Results
		More Recent Results
	Subsequence Recurrence
	Multiple Recurrence
	Connections with Combinatorics and Number Theory
	Future Directions
	References
Ergodic Theorems
	Glossary
	Definition of the Subject
	Introduction
	Ergodic Theorems for Measure-Preserving Maps
	Generalizations to Continuous Time and Higher-Dimensional Time
	Pointwise Ergodic Theorems for Operators
	Subadditive and Multiplicative Ergodic Theorems
	Entropy and the Shannon-McMillan-Breiman Theorem
	Amenable Groups
	Subsequence and Weighted Theorems
	Ergodic Theorems and Multiple Recurrence
	Rates of Convergence
	Ergodic Theorems for Non-amenable Groups
	Future Directions
	Bibliography
Spectral Theory of Dynamical Systems
	Glossary and Notation
	Definition of the Subject
	Introduction
		General Unitary Representations
		Koopman Representations
		Markov Operators, Joinings and Koopman Representations, Disjointness and Spectral Disjointness, and Entropy
	Maximal Spectral Type of a Koopman Representation: Alexeyev´s Theorem
	Spectral Theory of Weighted Operators
		Maximal Spectral Type of Weighted Operators over Rotations
		The Multiplicity Problem for Weighted Operators over Rotations
		Remarks on the Banach Problem
		Lifting Mixing Properties
	The Multiplicity Function
		Cocycle Approach
		Multiplicity for Gaussian and Poissonian Automorphisms
		Rokhlin´s Uniform Multiplicity Problem
	Rokhlin Cocycles
	Rank-1 and Related Systems
	Spectral Theory of Dynamical Systems of Probabilistic Origin
	Inducing and Spectral Theory
	Rigid Sequences
	Spectral Theory of Parabolic Dynamical Systems
		Time-Changes of Algebraic Systems
		Special Flows, Flows on Surfaces, and Interval Exchange Transformations
			Interval Exchange Transformations
			Smooth Flows on Surfaces and Their Special Representations
		Special Flows over Rotations and Interval Exchange Transformations
			Bounded Variation Roof Function
			Symmetric Logarithmic Singularities
			Asymmetric Logarithmic Singularities
			Power Singularities
	Spectral Theory for Locally Compact Groups of Type I
		Groups of Type I
		Spectral Properties of Heisenberg Group Actions
		Heisenberg Odometers
		On the ``Finitely Dimensional´´ Part of the Spectrum
	Future Directions
	Bibliography
Joinings in Ergodic Theory
	Glossary
		Disjoint Measure-Preserving Systems
		Joining
		Marginal of a Probability Measure on a Product Space
		Markov Intertwining
		Minimal Self-Joinings
		Off-Diagonal Self-Joinings
		Process in a Measure-Preserving Dynamical Systems
		Self-Joining
		Simplicity
	Definition of the Subject
	Introduction
	Joinings of Two or More Dynamical Systems
		The Set of Joinings
		From Disjointness to Isomorphy
			Disjointness
			Joinings and Isomorphism
		Joinings and Factors
		Markov Intertwinings and Composition of Joinings
	Self-Joinings
		Self-Joinings and Commuting Transformations
		Minimal Self-Joinings
		Simple Systems
		Relative Properties with Respect to a Factor
	Some Applications and Future Directions
		Filtering Problems
		Joinings Proofs of Ornstein´s and Krieger´s Theorems
		Joinings and Rohlin´s Multifold-Mixing Question
			Pairwise-Independent Joinings
			Host´s and Ryzhikov´s Theorems
		Joinings and Multiple Ergodic Averages
		Joinings and Conjectures in Number Theory
		Future Directions
	Bibliography
Entropy in Ergodic Theory
	Glossary
	Definition of the Subject
	Entropy Example: How Many Questions?
	Distribution Entropy
		Function η
		Distropy Fact
		Binomial Coefficients
	A Glance at Shannon´s Noisy Channel Theorem
		Noisy Channel
			Encoding/Decoding
	The Information Function
		Conditioning on a Field
			Conditional-Distropy Fact
		Entropy of a Process
		Generators
		Time Reversibility
		Bernoulli Processes
	Entropy of a Transformation
		Entropy Is Continuous
		Entropy Is Not Continuous
			Further Results
		Meshalkin´s Map
			The Code
		Markov Shifts
	Determinism and Zero-Entropy
		Rotations Are Deterministic
			Counting Names in a Rotation
			Consequences
		Rank-1 Has Zero-Entropy
		Cautions on Determinism´s Relation to Zero-Entropy
	The Pinsker-Field and K-Automorphisms
		K-Processes
	Ornstein Theory
	Skew Product and Random Walk in Random Scenery
	Topological Entropy
		Using a Metric
		Metric Preliminaries
		You Take the High Road and I´ll Take the Low Road
			Pretension
			Topological Markov Shifts
			The Golden Shift
			Top-Ent of the Golden Shift
			Top-Ent of a General TMS
			Labeling Edges
		The Variational Principle
			Topology on
	Entropy of a Flow(-Action)
	Entropy of d-Actions
		Complexity of Zero Entropy Actions
	Finitely Observable Invariant
		Finitely Observable Extension
	Recent Progress
		Entropy of Actions of Nonamenable Groups
		Weak Pinsker Conjecture
	Exodos
	Bibliography
		Articles, Papers, and Books of Interest
Isomorphism Theory in Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction
	Basic Transformations
	Basic Isomorphism Invariants
	Basic Tools
	Isomorphism of Bernoulli Shifts
		Kolmogorov-Sinai
		Explicit Isomorphisms
		Ornstein
		Properties of Bernoullis
		Rudolph Structure Theorem
	Transformations Isomorphic to Bernoulli Shifts
	Transformations Not Isomorphic to Bernoulli Shifts
		Rudolph´s Counterexample Machine
		T, T Inverse
	Classifying the Invariant Measures of Algebraic Actions
	Finitary Isomorphisms
	Flows
	Other Equivalence Relations
		Kakutani Equivalence
		Restricted Orbit Equivalence
	Non-invertible Transformations
		Markov Shifts
		Rational Maps
		Differences with Ornstein´s Theory
	Factors of a Transformation
	Actions of Amenable Groups
		Differences Between Actions of  and Actions of Other Groups
	Future Directions
	Bibliography
Dynamical Systems of Probabilistic Origin: Gaussian and Poisson Systems
	Glossary
	Definition of the Subject
	Introduction
	From Probabilistic Objects to Dynamical Systems
		Gaussian Systems
		Poisson Suspensions
	Spectral Theory
		Basics of Spectral Theory
		Fock Space
		Operators on a Fock Space
		Application to Gaussian and Poisson Chaos
			Fock Space Structure of L2 for Gaussian Dynamical Systems and Poisson Suspensions
			Second Quantization Operators
	Basic Ergodic Properties
		Ergodicity and Mixing
		Entropy, Bernoulli Properties
	Joinings, Factors, and Centralizer
		Gaussian Factors and Centralizer
		Poisson Factors and Centralizer
		Gaussian and Poisson Self-Joinings
	GAGs and PAPs
		From Foias-Stratila to GAGs
		Poissonian Analog of Foias-Stratila Theorem and PAPs
		Properties of GAGs and PAPs
	Future Directions
	Bibliography
Ergodic Theory: Nonsingular Transformations
	Glossary
	Definition of the Subject
	Introduction and Basic Results
		Nonsingular Transformations
		Basic Properties of Conservativity and Ergodicity
		Mean and Pointwise Ergodic Theorems. Rokhlin Lemma
		Ergodic Decomposition
		Generators
		The Glimm-Effros Theorem
		Minimal Radon Uniquely Ergodic Models for Infinite Measure-Preserving Transformations
		Special Representations of Ergodic Flows
	Panorama of Examples
		Nonsingular Product Odometers
		Markov Odometers
		Tower Transformations
		Rank-One Transformations. Chacón Maps. Finite Rank
		Nonsingular Bernoulli Shifts
		Nonsingular Markov Shifts
		Nonsingular Poisson Suspensions
		Nonsingular Gaussian Transformations
		IDPFT Transformations
		Natural Extensions of Nonsingular Endomorphisms
	Topological Groups U(X,μ), U2(X,μ)andAUT1(X,μ)
	Orbit Theory
		Full Groups. Ratio Set and Types IIIλ, 0  λ  1
		Maharam Extension, Associated Flow, and Orbit Classification of Type III Systems
		Almost Continuous Orbit Equivalence
		Normalizer of the Full Group. Outer Conjugacy Problem
		Cocycles of Dynamical Systems. Weak Equivalence of Cocycles
		ITPFI Transformations and AT-Flows
	Mixing Notions and Multiple Recurrence
		Weak Mixing
		Rational Ergodicity and Rational Weak Mixing
		Mixing, Zero-Type
		K-property
		Multiple and Polynomial Recurrence
	Dynamical Properties of IDPFT Systems
	Dynamical Properties of Nonsingular Bernoulli and Markov Shifts
		Krengel Class
		Generalized Krengel Class
		General Nonsingular Bernoulli Shifts
		Bernoulli Factors of Nonsingular Bernoulli Shifts
		Nonsingular Markov Shifts
	Dynamical Properties of Nonsingular Poisson Suspensions and Nonsingular Gaussian Transformations
		Nonsingular Poisson Suspensions
		Nonsingular Gaussian Transformations
	Spectral Theory for Nonsingular Systems
		L - spectrum and Groups of Quasi-invariance
		Koopman Unitary Operator for a Nonsingular System
		Koopman Unitary Operators Associated with Nonsingular Poisson Transformations
		Koopman Unitary Operators Associated with Nonsingular Gaussian Transformations
	Entropy and Other Invariants
		Krengel and Parry´s Entropies
		Poisson Entropy
		Parry´s Generalization of Shannon-MacMillan-Breiman Theorem
		Critical Dimension
		Nonsingular Restricted Orbit Equivalence
	Nonsingular Joinings and Factors
		Joinings, Nonsingular MSJ, and Simplicity
		Nonsingular Coding and Factors of Cartesian Products of Nonsingular Maps
		Joinings and MSJ for Infinite Measure-Preserving Systems
	Smooth Nonsingular Transformations
	Miscellaneous Topics
		On Normalizing Constants for Ergodic Theorem
		Around King´s Weak Closure Theorem
		Asymmetry and Bergelson´s Question
		Ergodicity of Powers
		Rigidity Sequences
		Directional Recurrence
	Applications. Connections with Other Fields
		Mild Mixing
		Ergodicity of Gaussian Cocycles
		Disjointness and Furstenberg´s Class
		Symmetric Stable and Infinitely Divisible Stationary Processes
		Poisson Suspensions of Infinite Measure-Preserving Transformations
		Recurrence of Random Walks with Nonstationary Increments
		Boundaries of Random Walks
		Stationary Actions
		Classifying σ-finite Ergodic Invariant Measures
		Von Neumann Algebras
		Representations of CAR
		Unitary Representations of Locally Compact Groups
	Further Directions
	Bibliography
Sarnak´s Conjecture from the Ergodic Theory Point of View
	Glossary
	Definition of the Subject
	Introduction
	Chowla Conjecture
		(C)
		(C) vs. (Clog)
		More on (Clog)
		Averaged (C)
		(C) vs. Other Conjectures
	Sarnak´s Conjecture
		(S)
		(S) vs. (C)
		Strong MOMO Property
		Möbius orthogonality of Positive Entropy Systems
		(Slog) vs. (Clog)
		(S) vs. (Slog)
		Strategies
	Arithmetic Properties of the Möbius Function
		Multiplicativity
		Aperiodicity
		Behaviour on Short Intervals
		Logarithmic Furstenberg Systems
	Future Directions
		Detecting Zero Entropy
		Proving the Strong MOMO Property
		Mixing Properties of Furstenberg Systems
		Furstenberg Disjointness in Non-ergodic Case
	Bibliography
		Further Reading
Smooth Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction
	The Volume Class
	The Fundamental Questions
	Lebesgue Measure and Local Properties of Volume
	Ergodicity of the Basic Examples
	Hyperbolic Systems
		Examples of Hyperbolic Maps and Attractors
			Expanding Maps
			Anosov Diffeomorphisms
			DA Attractors
			Distortion Estimates
		Ergodicity of Expanding Maps
		Ergodicity of Conservative Anosov Diffeomorphisms
			The Hopf Argument
			Absolute Continuity
		SRB Measures
	Beyond Uniform Hyperbolicity
		Partial Hyperbolicity
		Conservative Partially Hyperbolic Diffeomorphisms
		Dissipative Partially Hyperbolic Diffeomorphisms
		Nonuniform Hyperbolicity
			Hyperbolic Blocks
			Ergodic Properties of Nonuniformly Hyperbolic Diffeomorphisms
			The Dissipative Case
	The Presence of Critical Points and Other Singularities
		Hyperbolic Billiards and Hard Sphere Gases
		Interval Maps and Parameter Exclusion
		Near-Critical Diffeomorphisms
	Future Directions
	Bibliography
Ergodic and Spectral Theory of Area-Preserving Flows on Surfaces
	Glossary
	Definition of the Subject
	Introduction
	Examples
	Locally Hamiltonian flows
	Background and Tools
	Invariant Measures and (Unique) Ergodicity
	Behavior of Ergodic Averages
	Mixing Properties
	Spectral Properties
	Disjointness Results
	Open Directions
		Locally Hamiltonian Flows
	Acknowledgments
	Bibliography
Pressure and Equilibrium States in Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction
	Warming Up: Thermodynamic Formalism for Finite Systems
		Equilibrium Distributions and the Gibbs Property
		Systems on a Finite Lattice
	Shift Spaces, Invariant Measures, and Entropy
		Symbolic Dynamics
		Invariant Measures
		Entropy of Invariant Measures
			Theorem (Shannon-McMillan-Breiman Theorem)
				Asymptotic Equipartition Property
		A Short Digression on Complexity
		Entropy as a Function of the Measure
	The Variational Principle: A Global Characterization of Equilibrium
		Equilibrium States
			Equilibrium States as (Sub)-gradients
		The Variational Principle
			Theorem (Variational Principle for the Pressure)
		Nonuniqueness of Equilibrium States: An Example
		More on Equilibrium States
	The Gibbs Property: A Local Characterization of Equilibrium
		Subshifts of Finite Type
		The Gibbs Property for a Class of Regular Potentials
			Ruelle´s Perron-Frobenius Theorem
		Relative Entropy
		More Properties of Gibbs States
	Examples on Shift Spaces
		Measure of Maximal Entropy and Periodic Points
		Markov Chains Over Finite Alphabets
		The Ising Chain
		More on Hofbauer´s Example
	Examples from Differentiable Dynamics
		Uniformly Expanding Markov Maps of the Interval
		Interval Maps with an Indifferent Fixed Point
		Axiom A Diffeomorphisms, Anosov Diffeomorphisms, Sinai-Ruelle-Bowen Measures
		Bowen´s Formula for the Hausdorff Dimension of Conformal Repellers
	Nonequilibrium Steady States and Entropy Production
	Some Ongoing Developments and Future Directions
	Bibliography
Parallels Between Topological Dynamics and Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction and History
	Recurrence and Other Dynamical Properties
		Minimality Versus Ergodicity
			Minimality and Transitivity
			Ergodicity
			Ergodic Decomposition
		Equicontinuity Versus Measurable Kronecker
			Kronecker Systems
			Equicontinuity
		Topological Mixing Versus Measurable Mixing
			Measurable Mixing
			Topological Mixing
		Topological Weak Mixing Versus Measurable Weak Mixing
			Topological Weak Mixing
			Measurable Weak Mixing
		Topological Mild Mixing Versus Measurable Mild Mixing
			Measurable Mild Mixing
			Topological Mild Mixing
		Ellis Semigroup
		Disjointness and Weak Disjointness
		Chaos and Complexity: Topological Versus Measurable
		Meet Together
	Entropy Theory
		Entropy: Topological Versus Measurable
		Measurable and Topological K-Systems
		Measurable and Topological Entropy Tuples
			Entropy N-Tuples
			Local Variational Principles - The Connection of Two Kinds of Tuples
			The Ergodic Decomposition
			Weak Horseshoe
		Sequence Entropy: Topological Versus Measurable
		Null Systems: Measurable Versus Topological
		Maximal Pattern Entropy
		Other Tuples
	Structure Theorems and Multiple Ergodic Averages
		Structure Theorems in Topological Dynamics
		Structure Theorems in Ergodic Theory
		Multiple Ergodic Averages
		Characteristic Factors in Ergodic Theory
		Characteristic Factors in Topological Dynamics
		Topological Methods in the Study of the Multiple Ergodic Averages
	Further Directions
		Pointwise Convergence of the Multiple Averages
		Analogue Results
		Ramsey Type Theorems
		Sarnak Conjecture
	References
Symbolic Dynamics
	Glossary
	Definition of the Subject
	Introduction
	Origins of Symbolic Dynamics: Modeling of Dynamical Systems
	Shift Spaces and Sliding Block Codes
	Shifts of Finite Type and Sofic Shifts
	Entropy and Periodic Points
	The Conjugacy Problem
	Other Coding Problems
	Coding for Data Recording Channels
	Connections with Information Theory and Ergodic Theory
	Higher Dimensional Shift Spaces
	Future Directions
	Addendum to the Second Edition
	Bibliography
Operator Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction
	The Mean Ergodic Theorem
	Rates of Convergence
	Uniform Ergodic Theorems
	Strong Cesàro Convergence
	Weak Stability and Mixing
	Stability
	Averaging Along Subsequences
	General Averaging Methods
	Modulated Ergodic Theorems
	Resolvent Conditions and Growth of Powers
	Continuous Time (C0-semigroups)
	Bibliography
		References
		Books
Dynamical Systems and C-Algebras
	Glossary
	Definition of the Subject
	Introduction
	Topological Orbit Equivalence
		Étale Equivalence Relations
		Invariants of Étale Equivalence Relations
		AF-Equivalence Relations and Bratteli-Vershik Transformations
		The Bratteli-Vershik Model
		The Classification up to Isomorphisms of AF-Equivalence Relations: The Bratteli-Elliott-Krieger Theorem
		The Absorption Theorem
		Orbit Equivalence of AF-Equivalence Relations
		Orbit Equivalence of Minimal Actions of a Finitely Generated Abelian Group
		A Topological Krieger Theorem: The Notion of Strong Orbit Equivalence
		Continuous Orbit Equivalence
		Continuous Orbit Equivalence and C-algebra
		d-odometers
		Cohomology of d-odometers
	Mean Dimension, Small Boundary Property, and Classification of C-Algebras
		Mean Dimension and Radius of Comparison
		Classification of Simple Nuclear C-algebras
		Almost Finiteness, Comparison, and the Small Boundary Property
	Boundary Actions and C-Simplicity
		The Furstenberg Boundary
		The Hamana Boundary
	C-Simplicity and Unique Trace Property
	References
The Complexity and the Structure and Classification of Dynamical Systems
	Glossary
	Introduction: What Is a Dynamical System? What Is Structure? What Is a Classification?
	Examples of Structure and Classification Results in Dynamical Systems
		Hamiltonian Dynamics
		Rotations of the Unit Circle
		Symbolic Shifts
		Measure Structure
			Rolling Dice: Bernoulli Shifts
			Symbolic Systems as Models for Measure-Preserving Transformations
			Koopman Operators
			Translations on Compact Groups
		Presentations
		Smooth Dynamics
		More Detailed Structure Theory
		Summary
	Descriptive Complexity
		What Is a Reduction?
			Reductions of sets
			Reductions of Equivalence Relations
			Continuous Reductions
			The Pre-ordering
		Reducing Ill-Founded Trees
	Complexity in Structure Theory
		Examples at Three Levels of Complexity
		Non-recursive Classes
		Borel Classes
		Natural Classes that Are Not Borel
	Complexity in Classification Theory
		Equivalence Relations with Countable Classes
		The Glimm-Effros Dichotomy: E0
		=+ and the Friedman-Stanley Jump Operator
		S- Actions
		Turbulence
		Polish Group Actions
	Standard Mathematical Objects in Each Region
		Countable Equivalence Relations
		General Polish Group Actions
		Borel Equivalence Relations that Don´t Arise from a Polish Group Actions
		The Maximal Analytic Equivalence Relation
	Placing Dynamical Systems in Each Region
		Measure Isomorphism
			Systems with Numerical Invariants
			Reducing E0 to Dynamical Systems
			A Class Bi-reducible to =+
			Isomorphism Is Not Reducible to Any S-Action.
			Isomorphism Is Not Borel
			Smooth Measure-Preserving Transformations
			Kakutani Equivalence
			The Summary Diagram
		Topological Conjugacy
			Smooth Transformations with Numerical Invariants
			Diffeomorphisms of the Torus Above E0.
			A Dynamical Class that Is  Maximal for Countable Equivalence Relations
			An Example of a Maximal S-Action from Dynamical Systems
			Topological Conjugacy of Diffeomorphisms Is Not Borel
			The Summary Diagram
		A Descriptive Set Theory Facts
		Reductions and Hierarchies
		Open Problems
	Bibliography
Ergodic Theory: Interactions with Combinatorics and Number Theory
	Glossary
	Definition of the Subject
	Introduction
	Ergodic Theory
	Frequency of Returns
		Normal Numbers
		Continued Fraction Digits
		First Digits
		Equidistribution
		The Ergodic Context
	Ergodic Ramsey Theory and Recurrence
		Topology and Coloring Theorems
		Polynomialization and IP-sets
		Sets of Primes
	Orbit-Counting as an Analogous Development
		Counting Orbits and Geodesics
		Counting Orbits for Group Endomorphisms
		Exotic Orbit Growth
		Pólya-Carlson Dichotomy
	Diophantine Analysis as a Toolbox
		Orbit Growth and Convergence
		Mixing and Additive Relations in Fields
	Future Directions
	Bibliography
		Primary Literature
		Books and Reviews
Ergodic Theory on Homogeneous Spaces and Metric Number Theory
	Glossary
	Definition of the Subject
	Introduction
	Basic Facts
		Connection with Dynamics on the Space of Lattices
	Further Results
		Diophantine Approximation with Dependent Quantities: The Set-Up
	Further Results
	Future Directions
	Acknowledgments
	Bibliography
		New List
Ergodic Theory: Rigidity
	Glossary
	Definition of the Subject
	Introduction
	Basic Definitions and Examples
	Differentiable Rigidity
	Local Rigidity
	Global Rigidity
	Measure Rigidity
	Future Directions
	Acknowledgments
	Bibliography
		Primary Literature
		Books and Reviews
Chaos and Ergodic Theory
	Glossary
	Definition of the Subject
	Introduction
		Attempts at Definition
		Elementary Chaos: A Simple Example
	Picking an Invariant Probability Measure
		Statistical Descriptions
			Birkhoff Pointwise Ergodic Theorem
		Physical Measures
		Measures of Maximum Entropy
		Other Points of View
	Tractable Chaotic Dynamics
		The Palis Conjecture
		Uniformly Expanding/Hyperbolic Systems
		Pesin Theory
		Systems with Discontinuities
		Interval Maps with Critical Points
		Non-uniform Expansion/Contraction
		Hénon-Like Maps and Rank One Attractors
	Statistical Properties
		Probabilistic Limit Theorems
		Almost Sure Results
		Other Statistical Properties
	Orbit Complexity
		The Variational Principle
		Strict Inequality
		Orbit Complexity on the Set of Measures
		Local Complexity
		Global Simplicity
	Stability
		Structural Stability
		Continuity Properties of the Topological Dynamics
		Statistical Stability
		Stochastic Stability
	Untreated Topics
	Future Directions
		General Theory
		Physical Measures
		Maximum Entropy Measures and Topological Complexity
	Acknowledgments
	Bibliography
		Primary Literature
		Books and Reviews
Ergodic Theory: Fractal Geometry
	Glossary
	Definition of the Subject
	Introduction
	Preliminaries
		Some Ergodic Theory
		Hausdorff Dimension
		Dimension of a Measure
		Pointwise Dimension
		Dimension-Like Characteristics and Topological Entropy
		The Pressure Functional
	Brief Tour Through Some Examples
		Dimension of Conformal Repellers: Ruelle´s Pressure Formula
		Iterated Function Systems
		Homoclinic Bifurcations for Dissipative Surface Diffeomorphisms
		Some Applications to Number Theory
		Infinite Iterated Function Systems and Parabolic Systems
		Complex Dynamics
		Embedology and Computational Aspects of Dimension
		Denjoy Systems
		Return Times and Dimension
	Dimension Theory of Low-Dimensional Dynamical Systems - Young´s Dimension Formula
		Some Remarks on Dimension Theory for Low-Dimensional Versus High-Dimensional Dynamical Systems
	Dimension Theory of Higher-Dimensional Dynamical Systems
	Hyperbolic Measures
		The Kaplan-Yorke Conjecture
	General Theory
		The Existence of the Pointwise Dimension for Hyperbolic Measure - The Eckmann-Ruelle Conjecture
	Endomorphisms
	Multifractal Analysis
		The Dynamical Characteristic View of Multifractal Analysis
		General Multifractal Formalism
		The Entropy Spectrum
		The Dimension Spectrum
		The Lyapunov Spectrum
		Multifractal Analysis and Large Deviation Theory
	Future Directions
	Bibliography
		Primary Literature
		Books and Reviews
Index