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This book discusses basic topics in the spectral theory of dynamical systems. It also includes two advanced theorems, one by H. Helson and W. Parry, and another by B. Host. Moreover, Ornstein's family of mixing rank-one automorphisms is given with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are also examined. Baire category theorems of ergodic theory, scattered in literature, are discussed in a unified way in the book. Riesz products are introduced and applied to describe the spectral types and eigenvalues of rank-one automorphisms. Lastly, the second edition includes a new chapter "Calculus of Generalized Riesz Products", which discusses the recent work connecting generalized Riesz products, Hardy classes, Banach's problem of simple Lebesgue spectrum in ergodic theory and flat polynomials.
List of contents
The Hahn-Hellinger Theorem.- The Spectral Theorem for Unitary Operators.- Symmetry and Denseness of the Spectrum.- Multiplicity and Rank.- The Skew Product.- A Theorem of Helson and Parry.- Probability Measures on the Circle Group.- Baire Category Theorems of Ergodic Theory.- Translations of Measures on the Circle.- B. Host's Theorem.- L Eigenvalues of Non-Singular Automorphisms.- Generalities on Systems of Imprimitivity.- Dual Systems of Imprimitivity.- Saturated Subgroups of the Circle Group.- Riesz Products As Spectral Measures.- Additional Topics.- Calculus of Generalized Riesz Products.
About the author
Mahendra Nadkarni is Emeritus Professor of mathematics at the University of Mumbai, India. He earned his M.Sc. degree in Mathematics from the University of Mumbai in 1960 and Ph.D. in applied mathematics from Brown University in1965. His research and pedagogic interests are in the areas of ergodic theory, harmonic analysis and probability theory in which he has scholarly publications including a book titled
Basic Ergodic Theory. He is Fellow of the Indian National Science Academy, New Delhi, and, the Indian Academy of Sciences, Bengaluru. He has taught at the Indian Statistical Institute, Kolkata; the University of Minnesota, Minneapolis; Chennai Mathematical Institute, Chennai; the Indian Institute of Technology Indore; the Indian Institute of Technology Kanpur; Center of Excellence in Basic Sciences, Mumbai; and had visiting appointments at McGill University, Universities of York and Toronto in Canada.
