## Illinois Journal of Mathematics

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

#### 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

https://projecteuclid.org/euclid.ijm/1258138165

Digital Object Identifier
doi:10.1215/ijm/1258138165

Mathematical Reviews number (MathSciNet)
MR1879249

Zentralblatt MATH identifier
1055.37013

#### 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