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Spectral Theory of Dynamical Systems

Second Edition

  • Book
  • © 2020

Overview

Authors:
  1. Mahendra Nadkarni

    1. Emeritus Professor, Department of Mathematics, University of Mumbai, Mumbai, India

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  • Discusses major topics in the spectral theory of dynamical systems
  • Includes two advanced theorems: a theorem by Helson and Parry, and B. Host’s theorem
  • Features a new chapter on “Calculus of Generalized Riesz Products”
  • Describes Ornstein’s family of mixing rank-one automorphisms with construction and proof
  • Explains systems of imprimitivity and their relevance to ergodic theory, as well as Baire category theorems

Part of the book series: Texts and Readings in Mathematics (TRIM, volume 15)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xiii
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  2. The Hahn-Hellinger Theorem

    • Mahendra Nadkarni
    Pages 1-17

  3. The Spectral Theorem for Unitary Operators

    • Mahendra Nadkarni
    Pages 19-21

  4. Symmetry and Denseness of the Spectrum

    • Mahendra Nadkarni
    Pages 23-28

  5. Multiplicity and Rank

    • Mahendra Nadkarni
    Pages 29-33

  6. The Skew Product

    • Mahendra Nadkarni
    Pages 35-37

  7. A Theorem of Helson and Parry

    • Mahendra Nadkarni
    Pages 39-46

  8. Probability Measures on the Circle Group

    • Mahendra Nadkarni
    Pages 47-53

  9. Baire Category Theorems of Ergodic Theory

    • Mahendra Nadkarni
    Pages 55-65

  10. Translations of Measures on the Circle

    • Mahendra Nadkarni
    Pages 67-77

  11. B. Host’s Theorem

    • Mahendra Nadkarni
    Pages 79-86

  12. L∞ Eigenvalues of Non-Singular Automorphisms

    • Mahendra Nadkarni
    Pages 87-93

  13. Generalities on Systems of Imprimitivity

    • Mahendra Nadkarni
    Pages 95-102

  14. Dual Systems of Imprimitivity

    • Mahendra Nadkarni
    Pages 103-112

  15. Saturated Subgroups of the Circle Group

    • Mahendra Nadkarni
    Pages 113-121

  16. Riesz Products As Spectral Measures

    • Mahendra Nadkarni
    Pages 123-153

  17. Additional Topics

    • Mahendra Nadkarni
    Pages 155-176

  18. Calculus of Generalized Riesz Products

    • Mahendra Nadkarni
    Pages 177-207

  19. Back Matter

    Pages 209-223
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Keywords

About this book

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.

Authors and Affiliations

  • Emeritus Professor, Department of Mathematics, University of Mumbai, Mumbai, India

    Mahendra Nadkarni

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. 

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