Ergodic Ip Polynomial Szemeredi Theorem
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Ergodic IP Polynomial Szemeredi Theorem
Proves a polynomial multiple recurrence theorem for finitely, many commuting, measure-preserving transformations of a probability space, extending a polynomial Szemeredi theorem. Several applications to the structure of recurrence in ergodic theory are given, some of which involve weakly mixing systems, for which the authors also prove a multiparameter weakly mixing polynomial ergodic theorem. Techniques and apparatus employed include a polynomialization of an IP structure theory, an extension of Hindman's theorem due to Milliken and Taylor, a polynomial version of the Hales-Jewett coloring theorem, and a theorem concerning limits of polynomially generated IP systems of unitary operators. Author information is not given. Annotation copyrighted by Book News, Inc., Portland, OR.
An Ergodic IP Polynomial Szemeredi Theorem
Author: Vitaly Bergelson
language: en
Publisher: American Mathematical Soc.
Release Date: 2000
The authors 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]).
Nilpotent Structures in Ergodic Theory
Author: Bernard Host
language: en
Publisher: American Mathematical Soc.
Release Date: 2018-12-12
Nilsystems play a key role in the structure theory of measure preserving systems, arising as the natural objects that describe the behavior of multiple ergodic averages. This book is a comprehensive treatment of their role in ergodic theory, covering development of the abstract theory leading to the structural statements, applications of these results, and connections to other fields. Starting with a summary of the relevant dynamical background, the book methodically develops the theory of cubic structures that give rise to nilpotent groups and reviews results on nilsystems and their properties that are scattered throughout the literature. These basic ingredients lay the groundwork for the ergodic structure theorems, and the book includes numerous formulations of these deep results, along with detailed proofs. The structure theorems have many applications, both in ergodic theory and in related fields; the book develops the connections to topological dynamics, combinatorics, and number theory, including an overview of the role of nilsystems in each of these areas. The final section is devoted to applications of the structure theory, covering numerous convergence and recurrence results. The book is aimed at graduate students and researchers in ergodic theory, along with those who work in the related areas of arithmetic combinatorics, harmonic analysis, and number theory.