Sequences of integers and ergodic transformations
type: Item
instance of: scholarly article
title: Sequences of integers and ergodic transformations
zbMATH Open document ID: 0672.28006
DOI: 10.1016/0001-8708(89)90071-6
author: Stanley Eigen
author: Arshag Hajian
published in: Advances in Mathematics
publication date: 1989-01-01T00:00:00Z
review text: A class of infinite measure preserving transformations is constructed related to direct sum decompositions of the natural numbers \({\mathbb{N}}\). Specifically, let \({\mathbb{V}}\) and \({\mathbb{W}}\) be infinite subsets of \({\mathbb{N}}\) such that each \(n\in {\mathbb{N}}\) is a unique sum \(n=v+w,\) with \(v\in {\mathbb{V}}\) and \(w\in {\mathbb{W}}.\) Then an ergodic infinite measure preserving transformation T is constructed with a special set A and a distinguished point x, such that: \(\mu (A)=1;\) the set A is exhaustive, weakly wandering for T under the sequence \({\mathbb{W}}\); and the return times of the point x to A is the sequence \({\mathbb{V}}\). It then follows that for almost every point \(a\in A,\) the return times of a to A, \({\mathbb{R}}(a)\) is a direct complement of \({\mathbb{W}}\) in the integers \({\mathbb{Z}}\), i.e. every \(z\in {\mathbb{Z}}\) is a unique sum, \(z=r\oplus w\) for \(r\in {\mathbb{R}}(a)\) and \(w\in {\mathbb{W}}.\)
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 4099608
zbMATH Keywords: exhaustive weakly wandering sequence
zbMATH Keywords: ergodic infinite measure preserving transformation
zbMATH Keywords: return times
MaRDI profile type: Publication
full work available at URL: 0001-8708(89)90071-6
OpenAlex ID: W1998438026
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description: scientific article; zbMATH DE number 4099608