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

TL;DR

This paper constructs examples of infinite measure-preserving ergodic systems that are ergodic with isometric coefficients but not weakly doubly ergodic, expanding understanding of their properties and relationships.

Contribution

It demonstrates the existence of systems with specific ergodic properties that were previously unknown, particularly those that are ergodic with isometric coefficients but not weakly doubly ergodic.

Findings

Existence of infinite measure-preserving transformations that are ergodic with isometric coefficients but not weakly doubly ergodic.

Construction of type III_λ examples of such systems.

Example of a topological dynamical system that is topologically rigid but not measure-theoretically rigid.

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 $\text{III}_\lambda$ examples of such systems, $0<\lambda\leq 1$. 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 $(n_i)$ that is not measure theoretically rigid along $(n_i)$ for any nonsingular ergodic finite measure.