On \(\mu\)-recurrent nonsingular endomorphisms
type: Item
instance of: scholarly article
title: On \(\mu\)-recurrent nonsingular endomorphisms
zbMATH Open document ID: 0645.28010
DOI: 10.1007/BF02776298
author: Cesar E. Silva
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let T be a non-singular invertible transformation acting on a \(\sigma\)- finite measure space (X,\(\mu)\). \textit{D. Maharam} [Fundam. Math. 56, 35-50 (1964; Zbl 0133.003)] introduced a skew product \(T^*: X\times {\mathbb{R}}^+\to X\times {\mathbb{R}}^+\) defined by \(T^*(x,y)=(Tx,y/w(x))\) where \(w(x)=\frac{d\mu T}{d\mu}(x)\). She showed that T is conservative if and only if \(T^*\) is conservative. The main result of this paper generalizes the above to non-invertible transformations. The author uses the idea of \(\mu\)-recurrence of T: T is \(\mu\) recurrent if for every measurable \(g\geq 0\), \(\sum_{i\geq 0}g(T^ ix)w_ i(x)\) only takes the values 0 or \(\infty\) a.e. (where \(w_ i(x)=w(x)...w(T^{i-1})).\) A non- singular endomorphism of a \(\sigma\)-finite space is then shown to be \(\mu\)-recurrent if and only if \(T^*\) is conservative. An extension of a theorem of \textit{W. Krieger} [Math. Ann. 223, 19-70 (1976; Zbl 0332.46045)] on the recurrence of the Radon-Nikodým derivatives of conservative non-singular automorphisms is given, followed by an application of the above to the construction of conservative ergodic invertible natural extensions for \(\mu\)-recurrent ergodic nonsingular endomorphisms.
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4051946
zbMATH Keywords: non-invertible transformations
zbMATH Keywords: non-singular endomorphism
zbMATH Keywords: Radon- Nikodým derivatives
zbMATH Keywords: conservative non-singular automorphisms
zbMATH Keywords: \(\mu \)- recurrent ergodic nonsingular endomorphisms
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
cites work: Q4157390
cites work: Inverse Limits of Measure Spaces
cites work: Q5793543
cites work: Q5673127
cites work: Q3926822
cites work: Q5827369
cites work: Ergodic theorems. With a supplement by Antoine Brunel
cites work: On ergodic flows and the isomorphism of factors
cites work: Incompressible transformations
cites work: Ergodicity of some mappings of the circle and the line
cites work: Ergodic and Spectral Analysis of Certain Infinite Measure Preserving Transformations
cites work: Q5533878
cites work: Exact endomorphisms of a Lebesgue space
cites work: Q3853865
cites work: On recurrence
cites work: Note on an ergodic theorem
full work available at URL: https://doi.org/10.1007/bf02776298
OpenAlex ID: W2034742190
description: scientific article; zbMATH DE number 4051946