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An Ergodic IP Polynomial Szemerédi Theorem
 
Vitaly Bergelson Ohio State University, Columbus, OH
Randall McCutcheon University of Maryland, College Park, MD
Front Cover for An Ergodic IP Polynomial Szemeredi Theorem
Available Formats:
Electronic ISBN: 978-1-4704-0286-0
Product Code: MEMO/146/695.E
106 pp 
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
Front Cover for An Ergodic IP Polynomial Szemeredi Theorem
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  • Front Cover for An Ergodic IP Polynomial Szemeredi Theorem
  • Back Cover for An Ergodic IP Polynomial Szemeredi Theorem
An Ergodic IP Polynomial Szemerédi Theorem
Vitaly Bergelson Ohio State University, Columbus, OH
Randall McCutcheon University of Maryland, College Park, MD
Available Formats:
Electronic ISBN:  978-1-4704-0286-0
Product Code:  MEMO/146/695.E
106 pp 
List Price: $52.00
MAA Member Price: $46.80
AMS Member Price: $31.20
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1462000
    MSC: Primary 28; Secondary 05; 11;

    We prove a polynomial multiple recurrence theorem for finitely many commuting measure preserving transformations of a probability space, extending a polynomial Szemerédi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IP-Szemerédi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus employed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polynomial version of the Hales-Jewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IP-systems of unitary operators ([BFM]).

    Readership

    Researchers interested in measure-preserving transformations, partitions of integers, Ramsey theory, sequences and sets.

  • Table of Contents
     
     
    • Chapters
    • 0. Introduction
    • 1. Formulation of main theorem
    • 2. Preliminaries
    • 3. Primitive extensions
    • 4. Relative polynomial mixing
    • 5. Completion of the proof
    • 6. Measure-theoretic applications
    • 7. Combinatorial applications
    • 8. For future investigation
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Volume: 1462000
MSC: Primary 28; Secondary 05; 11;

We prove a polynomial multiple recurrence theorem for finitely many commuting measure preserving transformations of a probability space, extending a polynomial Szemerédi theorem appearing in [BL1]. The linear case is a consequence of an ergodic IP-Szemerédi theorem of Furstenberg and Katznelson ([FK2]). Several applications to the fine structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which we also prove a multiparameter weakly mixing polynomial ergodic theorem. The techniques and apparatus employed include a polynomialization of an IP structure theory developed in [FK2], an extension of Hindman's theorem due to Milliken and Taylor ([M], [T]), a polynomial version of the Hales-Jewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IP-systems of unitary operators ([BFM]).

Readership

Researchers interested in measure-preserving transformations, partitions of integers, Ramsey theory, sequences and sets.

  • Chapters
  • 0. Introduction
  • 1. Formulation of main theorem
  • 2. Preliminaries
  • 3. Primitive extensions
  • 4. Relative polynomial mixing
  • 5. Completion of the proof
  • 6. Measure-theoretic applications
  • 7. Combinatorial applications
  • 8. For future investigation
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