A note on the ergodic theorem
type: Item
instance of: scholarly article
title: A note on the ergodic theorem
zbMATH Open document ID: 1347.37011
author: Maria Pires de Carvalho
author: Fernando Jorge Moreira
published in: Qualitative Theory of Dynamical Systems
publication date: 2015-01-27T00:00:00Z
review text: The authors give a demonstration of the Birkhoff Ergodic Theorem for ``half-invariant'' measures. They consider a measure space \((X,\mathcal B,\mu)\) and a map \(T:X\to X\) such that the measure \(\mu\) is half \(T\)-invariant, i.e., \(\mu(A)\geq\mu(T^{-1}(A))\), for any \(A\in\mathcal B\). The invariance of the measures allows to estimate frequency of visits to a set \(A\) since almost every point in the phase space is recurrent. Thanks to the half invariance, a composition operator \(\mathcal U_T(\varphi)=\varphi\circ T\) can be defined on \(L^1(X,\mu)\) which is a positive contraction in \(L^1(X,\mu)\). This operator induces on \(X\) a Hopf decomposition. The proof of the theorem is essentially obtained by estimating the points for which the convergence of the Birkhoff averages is too slow.
Mathematics Subject Classification ID: 37A30
Mathematics Subject Classification ID: 28D99
Mathematics Subject Classification ID: 37B20
zbMATH DE Number: 6391424
zbMATH Keywords: Birkhoff's averages
zbMATH Keywords: pointwise convergence
reviewed by: Alejandro Mesón
MaRDI profile type: Publication
full work available at URL: s12346-014-0116-x
OpenAlex ID: W1998959734
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DOI: 10.1007/S12346-014-0116-X
description: scientific article; zbMATH DE number 6391424