Spectral Theory of Dynamical Systems
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Spectral Theory of Dynamical Systems

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308 g
235x155x mm
Birkhäuser Advanced Texts Basler Lehrbücher

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1 The Hahn-Hellinger Theorem.- 2 The Spectral Theorem for Unitary Operators.- 3 Symmetry and Denseness of the Spectrum.- 4 Multiplicity and Rank.- 5 The Skew Product.- 6 A Theorem of Helson and Parry.- 7 Probability Measures on the Circle Group.- 8 Baire Category Theorems of Ergodic Theory.- 9 Translations of Measures on the Circle.- 10 B. Host's Theorem.- 11 L? Eigenvalues of Non-Singular Automorphisms.- 12 Generalities on Systems of Imprimitivity.- 13 Dual Systems of Imprimitivity.- 14 Saturated Subgroups of the Circle Group.- 15 Riesz Products As Spectral Measures.- 16 Additional Topics.- References.
This book treats some basic topics in the spectral theory of dynamical 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 ofimprimitivity are discussed. and they are used to describe Riesz products, suitably generalised, are considered 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. Iam deeply indebted to G. R. Goodson. He has edited the book and suggested a number of corrections and improvements in both content and language.