On the structure of quasi-invariant measures for non-discrete subgroups of \(\text{Diff}^{\omega}(S^{1})\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1331.37011
author: Julio C. Rebelo
publication date: 2013-10-23T00:00:00Z
Mathematics Subject Classification ID: 37A50
Mathematics Subject Classification ID: 60G30
Mathematics Subject Classification ID: 22F10
Mathematics Subject Classification ID: 37E10
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 37F35
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 6218500
zbMATH Keywords: probability measure
zbMATH Keywords: absolute continuity
zbMATH Keywords: quasi-invariant measures
zbMATH Keywords: non-discrete group
zbMATH Keywords: diffeomorphisms of circle
zbMATH Keywords: singular quasi-invariant measures
zbMATH Keywords: d-quasiconformal measure
zbMATH Keywords: diffeomorphisms group
MaRDI profile type: Publication
full work available at URL: pdt002
OpenAlex ID: W2015965707
DOI: 10.1112/PLMS/PDT002
title: On the structure of quasi-invariant measures for non-discrete subgroups of \(\text{Diff}^{\omega}(S^{1})\)
published in: Proceedings of the London Mathematical Society. Third Series
review text: Consider a group \(G\) of diffeomorphisms acting on the circle \(S^1\). A probability measure \(\mu\) on \(S^1\) is said to be quasi-invariant by \(G\) if, for every element \(g\in G\), the measure \(g^*\mu\) has the same null-sets as \(\mu\). Recall that a group \(G\subset\mathrm{Diff}^{\omega}(S^1)\) is said to be non-discrete if it contains a sequence of elements \(\{g_n\} \subset G\) converging to the identity (say in the \(C^\infty\)-topology) and such that \(g_n \neq\mathrm{id}\). The purpose of this paper is to study the nature of quasi-invariant measures \(\mu\) for finitely generated non-discrete subgroups of \(\mathrm{Diff}^{\omega}(S^1)\). In this paper the author proves some interesting results concerning an absolute continuity of the \(\mu\). In particular the following is proved. Let \(G\subset \mathrm{Diff}^{\omega}(S^1)\) be a finitely generated non-solvable and non-discrete group. Assume also that \(G\) has no finite orbit and that \(\mu\) is a probability measure quasi-invariant under \(G\). Then \(\mu\) is absolutely continuous. Concerning singular quasi-invariant measures, it is also proved that their associated Hausdorff measures must either be zero or of infinite mass, a result contrasting with the case of dynamically defined Cantor sets and also applicable to the examples of singular stationary measures constructed by Kaimanovich and Le Prince. As a further application of this method, a rigidity theorem for measurable conjugates between groups as above is obtained. The author notices that most results are valid in the \(C^\infty\) category and, in fact, part of them works also for diffeomorphisms of class \(C^2\).
reviewed by: Nicolai K. Smolentsev
description: scientific article; zbMATH DE number 6218500