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 (T n(A) A) > 0 and (T n(A) B) > 0. While double ergodicity is equivalent to weak mixing for nite measure-preserving transformations, we show that this is not the case for in nite 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 in nite measure-preserving transformations, double ergodicity lies properly between weak mixing and ergodic Cartesian square. In addition we study some properties of double ergodicity. 2000 Mathematics Subject Classi cation. Primary 37A40. Secondary 28D yNew College of the University of South Florida, 5700 N. Tamiami Tr., Sarasota, FL 34243, USA. abowles@virtu.sar.usf.edu Harvard University, Cambridge, MA, 02138, USA. dkowsk@post.harvard.edu Swarthmore College, Swarthmore, PA 19081, USA. aem@sccs.swarthmore.edu {Department of Mathematics, Williams College, Williamstown, MA 01267, USA. csilva@williams.edu