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