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. We also give type IIIλ examples of such systems, 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.
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