Electronic ISBN:  9781470402860 
Product Code:  MEMO/146/695.E 
106 pp 
List Price:  $52.00 
MAA Member Price:  $46.80 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 146; 2000MSC: 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 IPSzemerédi 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 HalesJewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IPsystems of unitary operators ([BFM]).
ReadershipResearchers interested in measurepreserving 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. Measuretheoretic applications

7. Combinatorial applications

8. For future investigation


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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 IPSzemerédi 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 HalesJewett coloring theorem ([BL2]), and a theorem concerning limits of polynomially generated IPsystems of unitary operators ([BFM]).
Researchers interested in measurepreserving 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. Measuretheoretic applications

7. Combinatorial applications

8. For future investigation