Disjointness between bounded rank-one transformations

TL;DR

This paper provides conditions under which bounded rank-one transformations are isomorphic or disjoint, and explores their self-joining properties, with implications for ergodic theory.

Contribution

It offers new criteria for isomorphism, disjointness, and minimal self-joinings of bounded rank-one transformations based on their parameters.

Findings

Conditions for isomorphism and disjointness based on parameters.

Complete characterization for commensurate, canonically bounded transformations.

Proof that certain transformations have minimal self-joinings of all orders.

Abstract

In this paper some sufficient conditions are given for when two bounded rank-one transformations are isomorphic or disjoint. For commensurate, canonically bounded rank-one transformations, isomorphism and disjointness are completely determined by simple conditions in terms of their cutting and spacer parameters. We also obtain sufficient conditions for bounded rank-one transformations to have minimal self-joinings. As an application, we give a proof of Ryzhikov's theorem that totally ergodic, non-rigid, bounded rank-one transformations have minimal self-joinings of all orders.