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.