Table Of ContentTEXTS AND READINGS 15
IN MATHEMATICS
Spectral Theory of
Dynamical Systems
Texts and Readings in Mathematics
Advisory Editor
C. S. Seshadri, Chennai Mathematical Institute, Chennai.
Managing Editor
Rajendra Bhatia, Indian Statistical Institute, New Delhi.
Editors
R. B. Bapat, Indian Statistical Institute, New Delhi.
V. S. Borkar, Tata Inst. of Fundamental Research, Mumbai.
Probal Chaudhuri, Indian Statistical Institute, Kolkata.
V. S. Sunder, Inst. of Mathematical Sciences, Chennai.
M. Vanninathan, TIFR Centre, Bangalore.
Spectral Theory of
Dynamical Systems
M.G. Nadkarni
University of Mumbai
Mumbai
[ldgl@OOHINDUSTAN
U ULJ UB OOK AGENCY
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ISBN 978-81-85931-17-3 ISBN 978-93-80250-93-9 (eBook)
DOI 10.1007/978-93-80250-93-9
Copyright © 1998, Hindustan Book Agency (India)
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Contents
Preface
1 The Hahn-Hellinger Theorem 1
Definitions and the Problem . . . . 1
The Case of Multiplicity One, Cyclic Vector· . . . 2
Application to Second Order Stochastic Processes 5
Spectral Measures of Higher Multiplicity: A Canonical
Example ................... . 8
Linear Operators Commuting with Multiplication 9
Spectral Type; Maximal Spectral Type . . . . 13
The Hahn-Hellinger Theorem (First Form) .' .. . 15
The Hahn-Hellinger Theorem (Second Form) .. . 17
Representation of Second Order Stochastic Processes 20
2 The Spectral Theorem for Unitary Operators 22
The Spectral Theorem: Multiplicity One Case . . 22
The Spectral Theorem: Higher Multiplicity Case 23
3 Symmetry and Denseness of the Spectrum 25
Spectrum UT: It is Symmetric. 25
Spectrum of UT: It is Dense 26
Examples ......... . 28
4 Multiplicity and Rank 31
A Theorem on Multiplicity . 31
Approximation with Multiplicity N 32
Rank and Multiplicity ...... . 34
5 The Skew Product 37
The Skew Product: Definition and its Measure Preserving
Property . . . . . . . . . . . 37
The Skew Product: Its Spectrum ......... , . . . .. 38
vi Contents
6 A Theorem of Helson and Parry 40
Statement of the Theorem . . . . . . 40
Weak von Neumann Automorphisms and Hyperfinite Actions 40
The Co cycle C(g, x). . . . . . . . . . . . . . . 41
The Random Co cycle and the Main Theorem 43
Remarks. . . . . . . . . . . . . . . . . . . . . 47
7 Probability Measures on the Circle Group 49
Continuous Probability Measures on S1: They are Dense Ga 49
Measures Orthogonal to a Given Measure ..... 51
Measures Singular Under Convolution And Folding 52
Rigid Measures . . . . . . . . . . . . . . . . . . . . 53
8 Baire Category Theorems of Ergodic Theory 57
Isometries of U(X,8,m) . . . . . . . . . 57
Strong Topology on Isometries . . . . . . . . . . 58
Coarse and Uniform Topologies on Q(m) . . . . 59
Baire Category of Classes of Unitary Operators 62
Baire Category of Classes of Non-Singular Automorphisms 66
Baire Category of Classes of Measure Preserving
Automorphisms . . . . 67
Baire Category and Joinings . . . . . . . . . 68
9 Translations of Measures on the Circle 71
A Theorem of Weil and Mackey . . . . . . . . . . . . . . . .. 71
The Sets A(J.t) and H(J.t) and Their Topologies. . . . . . . . . 74
Groups Generated by Dense Subsets of A(J.t); Their Properties 76
Ergodic Measures on the Circle Group 77
A Theorem on Marginal Measures. 81
10 B. Host's Theorem. 85
Pairwise Independent and Independent Joinings
of Automorphisms. . . . . . . . . . . . . . 85
B. Host's Theorem: The Statement ...... . 86
Mixing Implies Multiple Mixing if the Spectrum is Singular 87
B. Host's Theorem: The Proof 87
An Improvement and an Application . . . . . . . . . . . .. 92
Contents vi i
11 Loo Eigenvalues of Non-Singular Automorphisms 95
The Group of Eigenvalues and Its Polish Topology 95
Quasi-Invariance of the Spectrum 98
The Group e(T) is a-Compact. 99
The Group e(T) is Saturated. . . 100
12 Generalities on Systems of Imprimitivity 104
Spectral Measures and Group Actions 104
Cocycles; Systems of Imprimitivity . 107
Irreducible Systems of Imprimitivity 110
Transitive Systems ... 111
Transitive Systems on IR . . . . . . 111
13 Dual Systems of Imprimitivity 113
Compact Group Rotations; Dual Systems of Imprimitivity 113
Irreducible Dual Systems; Examples. . . . . . . . . . . .. 114
The Group of Quasi-Invariance; Its Topology .. , ... " 118
The Group of Quasi-Invariance; It is an Eigenvalue Group 119
Extensions of Cocycles . . . . . . . . . . . . . . . . . . .. 122
14 Saturated Subgroups of the Circle Group 125
Saturated Subgroups of SI . . . . . . . . . . . . 125
Relation to Closures and Convex Hulls of Characters 128
a-Compact Saturated Subgroups; H2 Groups 132
15 Riesz Products As Spectral Measures. 137
Dissociated Trigonometric Polynomials . . . . . . . . 137
Classical Riesz Products and a Theorem of Peyriere . 138
Riesz Products and Dynamics . . . . .. ...... 141
Generalised Riesz Products. . . . . . . . . . . . . . . 144
Maximal Spectral Types of Rank One Automorphisms 148
Examples and Remarks. . . . . . . . . . . . .. . . . . . 152
The Non-Singular Case, Proof of Theorem 15.18., and Fur-
ther Remarks ..................... 155
Rank One Automorphisms: Their Group of Eigenvalues 157
Preliminary Calculations . . . . . . . . . . 158
The Functions 'Yk. . . . . . . . . . . . . . . 159
The Eigenvalue Group: Osikawa Criterion 161
Restatement of Theorem 15.50. .... . . 163
viii Contents
The Eigenvalue Group: Structural Criterion. 164
4J:,
An Expression for et E e(T) . . .... . 170
16 Additional Topics. 174
Bounded Functions with Maximal Spectral Type 174
A Result on Mixing . . . . . . . . . . . . . 177
A Result On Multiplicity . . . . . . . . . . . 180
Combinatorial and Probabilistic Lemmas . . 182
Rank One Automorphisms by Construction 187
Ornstein's Class of Rank One Automorphisms 189
Mixing Rank One Automorphisms ...... . 191
References 201
Index 212
Preface
This book treats some basic topics in the spectral theory of dynami
cal systems, where by a dynamical system we mean a measure space on
which a group of automorphisms acts preserving the sets of measure zero.
The treatment is at a general level, but even here, two theorems which
are not on the surface, one due to H. Helson and W. Parry and the other
due to B. Host are presented. Moreover non-singular automorphisms
are considered and systems of imprimitivity are discussed. Riesz prod
ucts, suitably generalised, are considered and they are used to describe
the spectral types and eigenvalues of rank one automorphisms. On the
other hand topics such as spectral characterisations of various mixing
conditions, which can be found in most texts on ergodic theory, and
also the spectral theory of Gauss Dynamical Systems, which is very well
presented in Cornfeld, Fomin, and Sinai's book on Ergodic Theory, are
not treated in this book. A number of discussions and correspondence
on email with El Abdalaoui El Houcein made possible the presentation
of mixing rank one construction of D. S. Ornstein.
I am deeply indebted to G. R. Goodson. He has edited the book and
suggested a number of corrections and improvements in both content
and language.
M. G. Nadkarni
