Flows with uncountable but meager group of self-similarities
type: Item
instance of: scholarly article
title: Flows with uncountable but meager group of self-similarities
zbMATH Open document ID: 1276.37001
author: Alexandre I. Danilenko
published in: Dynamical Systems and Group Actions
publication date: 2012-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.2496
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 37A10
zbMATH DE Number: 6077498
zbMATH Keywords: ergodic flow
zbMATH Keywords: self-similarity
MaRDI profile type: Publication
OpenAlex ID: W1962254222
arXiv ID: 1108.2496
DOI: 10.1090/CONM/567/11231
review text: Let \((T_t)_{t\in\mathbb R}\) be an ergodic free measure-preserving flow on a standard probability space. Denote by \(T\circ s\) a flow \((T_{st})_{t\in\mathbb R}\). Let NEWLINE\[NEWLINEI(T)=\{s\in\mathbb R: s\neq0, \,T\circ s\text{ is isomorphic to }T\}.NEWLINE\]NEWLINE These are called self-similarities of \(T\). If \(I(T)\neq\{1,-1\}\), \(T\) is called self-similar. In this article, the author proves the following theorem: There exist weakly mixing Gaussian flows with uncountable but meager group of self-similarities. Given a probability measure \(\tau\) on a locally compact second countable abelian group, denote by \(H_G(\tau)\) the set of all \(g\in G\) such that the translation of \(\tau\) by \(g\) is equivalent to \(\tau\). Let NEWLINE\[NEWLINEI_{\mathrm{Po}}(T)=\{s\in\mathbb R: s\neq0,\, T\circ s\text{ is isomorphic to \(T\) via a Poisson transformation}\}.NEWLINE\]NEWLINE He also proves the following theorem: For each probability measure \(\kappa\) on \(\mathbb R_+\), there is a weakly mixing Poisson flow \(\tilde T\) with \(I_{\mathrm{Po}}(\tilde T)=H_{\mathbb R_+}(\kappa)\).NEWLINENEWLINEFor the entire collection see [Zbl 1237.37004].
reviewed by: Makoto Mori
description: scientific article; zbMATH DE number 6077498