Existence and non-existence of a finite invariant measure
type: Item
instance of: scholarly article
title: Existence and non-existence of a finite invariant measure
zbMATH Open document ID: 1266.37003
author: Arshag Hajian
author: Stanley Eigen
author: Yuji Ito
author: V. S. Prasad
published in: Tokyo Journal of Mathematics
publication date: 2013-03-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.tjm/1358951323
review text: This paper serves two distinct purposes. First, it collects into one place a list of seven conditions on a nonsingular transformation \(T\) on a finite measure space \(\{X, m\}\) which are known to be equivalent to the existence of a preserved measure \(\mu \sim m\). The authors provide a collection of techniques and results which are used to directly show their equivalence (Theorem 3.2). This section of the paper serves as a survey of prior results, rather than a new result in its own right. The second aim of the paper is the introduction of the notion of a ``strongly weakly wandering'' (sww) sequence, related to the existing notion of an ``exhaustive weakly wandering'' (eww) sequence. Whereas an (eww) sequence is associated to a set whose iterates under \(T\) given by the sequence are disjoint and cover \(X\), a (sww) sequence has no such covering requirement, albeit with a disjointness condition which more intricately depends on the arithmetic relation between terms of the sequence. The (sww) condition is shown to be equivalent to nonexistence of a preserved \(\mu \sim m\) (Propositions 4.5 and 4.6). The authors remark that few explicit examples exist of transformations with (eww) sequences, and that (sww) sequences are easier to construct, but do not include any such examples. Remarks indicate that a forthcoming monograph will contain further discussions, however.
reviewed by: David Ralston
Mathematics Subject Classification ID: 37A40
zbMATH DE Number: 6146574
zbMATH Keywords: recurrence
zbMATH Keywords: equivalent invariant measure
zbMATH Keywords: weakly wandering
zbMATH Keywords: exhaustive weakly wandering
zbMATH Keywords: strongly weakly wandering
MaRDI profile type: Publication
OpenAlex ID: W2019128307
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DOI: 10.3836/TJM/1358951323
description: scientific article; zbMATH DE number 6146574