Illinois Journal of Mathematics

Double ergodicity of nonsingular transformations and infinite measure-preserving staircase transformations

Amie Bowles, Lukasz Fidkowski, Amy E. Marinello, and Cesar E. Silva

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Abstract

A nonsingular transformation is said to be doubly ergodic if for all sets $A$ and $B$ of positive measure there exists an integer $n>0$ such that $\lambda(T^{-n}(A)\cap A)>0$ and $\lambda(T^{-n}(A)\cap B)>0$. While double ergodicity is equivalent to weak mixing for finite measure-preserving transformations, we show that this is not the case for infinite measure preserving transformations. We show that all measure-preserving tower staircase rank one constructions are doubly ergodic, but that there exist tower staircase transformations with non-ergodic Cartesian square. We also show that double ergodicity implies weak mixing but that there are weakly mixing skyscraper constructions that are not doubly ergodic. Thus, for infinite measure-preserving transformations, double ergodicity lies properly between weak mixing and ergodic Cartesian square. In addition we study some properties of double ergodicity.

Article information

Source
Illinois J. Math., Volume 45, Number 3 (2001), 999-1019.

Dates
First available in Project Euclid: 13 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258138165

Digital Object Identifier
doi:10.1215/ijm/1258138165

Mathematical Reviews number (MathSciNet)
MR1879249

Zentralblatt MATH identifier
1055.37013

Subjects
Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations
Secondary: 28D05: Measure-preserving transformations 37A25: Ergodicity, mixing, rates of mixing

Citation

Bowles, Amie; Fidkowski, Lukasz; Marinello, Amy E.; Silva, Cesar E. Double ergodicity of nonsingular transformations and infinite measure-preserving staircase transformations. Illinois J. Math. 45 (2001), no. 3, 999--1019. doi:10.1215/ijm/1258138165. https://projecteuclid.org/euclid.ijm/1258138165


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