Strict Doubly Ergodic Infinite Transformations

TL;DR

This paper constructs examples of rank-one infinite measure-preserving transformations that are weakly doubly ergodic and rigid but have non-ergodic 2-fold products, and explores conditions for their multiple Cartesian products to be conservative.

Contribution

It provides new examples and conditions for doubly ergodic infinite transformations and analyzes their properties in group actions and sensitivity.

Findings

Examples of transformations with specific ergodic properties.

Conditions for multiple Cartesian products to be conservative.

Connection between group actions, ergodicity, and sensitivity.

Abstract

We give examples of rank-one transformations that are (weak) doubly ergodic and rigid (so all their cartesian products are conservative), but with non-ergodic $2$-fold cartesian product. We give conditions for rank-one infinite measure-preserving transformations to be (weak) doubly ergodic and for their $k$-fold cartesian product to be conservative. We also show that a (weak) doubly ergodic nonsingular group action is ergodic with isometric coefficients, and that the latter strictly implies W measurable sensitivity.