Ergodic properties of some linear actions
type: Item
instance of: scholarly article
title: Ergodic properties of some linear actions
zbMATH Open document ID: 1004.37004
DOI: 10.1023/A:1011328016377
author: François Ledrappier
published in: Journal of Mathematical Sciences (New York)
publication date: 2002-10-30T00:00:00Z
review text: Let \(\Gamma\) be a discrete group of \((2\times 2)\)-matrices with determinant 1 such that the quotient \(\Gamma\setminus \text{SL}(2,\mathbb{R})\) is compact and let \(\rho:\Gamma\to \mathbb{Z}^d\) be a representation of \(\Gamma\) on \(\mathbb{Z}^d\). The author surveys some ergodic properties of the linear action of \(\overline{\Gamma}= \ker\rho\) on \(\mathbb{R}^2\). Let \(L\) be the Lebesgue measure on \(\mathbb{R}^2\). The author proves, among other things, the following: Assume \(-Id\in\Gamma\). Then there is a function \(A: (\mathbb{R}^2\setminus \{0\})\times \mathbb{R}^+\to \mathbb{R}^+\) such that for any continuous function \(f\) with compact support on \(\mathbb{R}^2\setminus \{0\}\), we have for \(L\)-almost every \(x\in \mathbb{R}^2\), \[ \lim_{T\to\infty} \Biggl( \sum_{\gamma\in \overline{\Gamma}, \|\gamma\|< T} f(Jx)/ A(x,T)\Biggr)= \int(f(y)/|y|) dL(y) \] and \[ \limsup_{T\to\infty} A(x,T) (\log T)^{d/2}/T= C|x|^{-1} \] for some constant \(C\). Moreover, the constant \(C\) is given by a precise formula. The proof goes through a precise estimate of the sum \(\sum_{\gamma\in \overline{\Gamma},\|\gamma\|< T} f(Jx)\) for sufficiently many functions \(f\).
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 37D35
Mathematics Subject Classification ID: 37D40
zbMATH DE Number: 1821783
zbMATH Keywords: linear actions
zbMATH Keywords: ergodicity
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
full work available at URL: a:1011328016377
OpenAlex ID: W135850803
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description: scientific article; zbMATH DE number 1821783