Spectral Theory of Dynamical Systems

Contents
Preface
Preface to the Second Edition
About the Author
Chapter 1
The Hahn-Hellinger Theorem
Definitions and the Problem
The Case of Multiplicity One, Cyclic Vector
Application to Second Order Stochastic Processes
Spectral Measures of Higher Multiplicity: A Canonical Example
Linear Operators Commuting with Multiplication
Spectral Type; Maximal Spectral Type
The Hahn-Hellinger Theorem (First Form)
The Hahn-Hellinger Theorem (Second Form)
Representation of Second Order Stochastic Processes
Chapter 2 The Spectral Theorem for Unitary Operators
The Spectral Theorem: Multiplicity One Case
The Spectral Theorem: Higher Multiplicity Case
Chapter 3 Symmetry and Denseness of the Spectrum
Spectrum of UT : It is Symmetric
Spectrum of UT : It is Dense
Examples
Chapter 4 Multiplicity and Rank
A Theorem on Multiplicity
Approximation with Multiplicity N
Rank and Multiplicity
Chapter 5 The Skew Product
The Skew Product: Definition and its Measure Preserving Property
The Skew Product: Its Spectrum
Chapter 6 A Theorem of Helson and Parry
Statement of the Theorem
Weak von Neumann Automorphisms and Hyperfinite Actions
The Random Cocycle and the Main Theorem
Remarks
Chapter 7 Probability Measures on the Circle Group
Continuous Probability Measures on S^1: They are Dense Gδ
Measures Orthogonal to a Given Measure
Measures Singular under Convolution and Folding
Rigid Measures
Chapter 8 Baire Category Theorems of Ergodic Theory
Isometries of Lp(X, B, m)
Strong Topology on Isometries
Coarse and Uniform Topologies on G(m)
Baire Category of Classes of Unitary Operators
Baire Category of Classes of Non-Singular Automorphisms
Baire Category of Classes of Measure Preserving Automorphisms
Baire Category and Joinings
Chapter 9 Translations of Measures on the Circle
A Theorem of Weil and Mackey
The Sets A(μ) and H(μ) and Their Topologies
Groups Generated by Dense Subsets of A(μ); Their Properties
Ergodic Measures on the Circle Group
A Theorem on Marginal Measures
Chapter 10 B. Host's Theorem
Pairwise Independent and Independent Joinings of Automorphisms
B. Host's Theorem: The Statement
Mixing Implies Multiple Mixing if the Spectrum is Singular
B. Host's Theorem: The Proof
Chapter 11 L^∞ Eigenvalues of Non-Singular Automorphisms
The Group of Eigenvalues and Its Polish Topology
The Group e(T ) is σ-compact
The Group e(T ) is Saturated
Chapter 12 Generalities on Systems of Imprimitivity
Spectral Measures and Group Actions
Cocycles; Systems of Imprimitivity
Irreducible Systems of Imprimitivity
Transitive Systems
Transitive Systems on R
Chapter 13 Dual Systems of Imprimitivity
Compact Group Rotations; Dual Systems of Imprimitivity
Irreducible Dual Systems; Examples
The Group of Quasi-Invariance; Its Topology
The Group of Quasi-Invariance; It is an Eigenvalue Group
Extensions of Cocycles
Chapter 14 Saturated Subgroups of the Circle Group
Saturated Subgroups of S^1
Relation to Closures and Convex Hulls of Characters
σ-Compact Saturated Subgroups; H2 Groups
Chapter 15 Riesz Products As Spectral Measures
Dissociated Trigonometric Polynomials
Classical Riesz Products and a Theorem of Peyriére
Riesz Products and Dynamics
Generalised Riesz Products
Maximal Spectral Types of Rank One Automorphisms
Examples and Remarks
The Non-Singular Case, Proof of Theorem 15.18, and Further Remarks
Rank One Automorphisms: Their Group of Eigenvalues
Preliminary Calculations
The Eigenvalue Group: Osikawa Criterion
Restatement of Theorem 15.50
The Eigenvalue Group: Structural Criterion
An Expression for dσα/dσ , α ∈ e(T)
Chapter 16 Additional Topics
Bounded Functions with Maximal Spectral Type
A Result on Mixing
A Result On Multiplicity
Combinatorial and Probabilistic Lemmas
Rank One Automorphisms by Construction
Ornstein's Class of Rank One Automorphisms
Mixing Rank One Automorphisms
Chapter 17 Calculus of Generalized Riesz Products
Generalized Riesz Products and their Weak Dichotomy
Outer Polynomials and Mahler Measure
A Formula for Radon Nikodym Derivative
A Conditional Strong Dichotomy and Other Discussion
Non-Singular Rank One Maps and Generalized Riesz Products
Generalized Riesz Products of Dynamical Origin
Zeros of Polynomials
References
Index