Nonsingular transformations that are ergodic with isometric coefficients and not weakly doubly ergodic

Abstract

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are ergodic with isometric coefficients but are not weakly doubly ergodic; hence these two notions are not equivalent for infinite measure. We also give type IIIλ examples of such systems, for 0<λ≤1. We prove that under certain hypotheses, systems that are weakly mixing are ergodic with isometric coefficients and along the way we give an example of a uniformly rigid topological dynamical system along the sequence (ni) that is not measure theoretically rigid along (ni) for any nonsingular ergodic finite measure.