Rank-one flows of transformations with infinite ergodic index

TL;DR

This paper constructs a specific type of infinite measure-preserving flow where each non-zero time transformation has ergodic Cartesian powers, advancing understanding of ergodic properties in infinite measure systems.

Contribution

It introduces a rank-one flow with infinite ergodic index, demonstrating ergodicity of all Cartesian powers of its non-zero time transformations, a novel example in ergodic theory.

Findings

Constructed a rank-one infinite measure-preserving flow with ergodic Cartesian powers.

Showed that for each non-zero time, the transformation's Cartesian powers are ergodic.

Provided a new example illustrating properties of infinite ergodic index flows.

Abstract

A rank-one infinite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that for each $t\ne 0$, the Cartesian powers of the transformation $T_t$ are all ergodic.