Topological multiple recurrence for polynomial configurations in nilpotent groups
type: Item
instance of: scholarly article
title: Topological multiple recurrence for polynomial configurations in nilpotent groups
zbMATH Open document ID: 1016.05070
DOI: 10.1016/S0001-8708(02)00052-X
published in: Advances in Mathematics
publication date: 2003-05-25T00:00:00Z
review text: Let \(G\) be a nilpotent group of self-homeomorphisms of a compact metric space \((X,\rho)\) and let \(P_1, \dots, P_k\) be polynomial mappings from the collection of finite sets of natural numbers to \(G\) satisfying \(P_1(\emptyset) = \cdots = P_k(\emptyset) = 1_G\). The main theorem of this paper shows that, for any \(\varepsilon > 0\), there exist \(x \in X\) and a nonempty finite set of natural numbers \(\alpha\) such that \(\rho(P_i(\alpha)x,x) < \varepsilon\) for all \(i = 1, \dots, k\). This theorem can be viewed as a nilpotent version of the authors' polynomial Hales-Jewett theorem [cf. Ann. Math. 150, 33-75 (1999; Zbl 0969.05061)] and contains nilpotent extensions of many known ``abelian'' results as special cases.
Mathematics Subject Classification ID: 05D10
Mathematics Subject Classification ID: 37B20
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1915638
zbMATH Keywords: polynomial Hales-Jewett theorem
zbMATH Keywords: Ramsey theory
zbMATH Keywords: topological dynamics
zbMATH Keywords: nilpotent group
author: V. Bergelson
author: A. Leibman
MaRDI profile type: Publication
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full work available at URL: s0001-8708(02)00052-x
OpenAlex ID: W1966794383
description: scientific article; zbMATH DE number 1915638