On Disjointness of Mixing Rank One Actions

TL;DR

This paper investigates the non-isomorphism of certain mixing rank-one flows and automorphisms, using joinings and construction parameters, highlighting the difficulty of classifying mixing flows under time changes.

Contribution

It introduces new criteria for non-isomorphism of mixing rank-one flows and automorphisms based on joinings and construction parameters.

Findings

Joinings technique proves non-isomorphism for mixing rank-one flows under linear time change.

Different construction parameters (height and cut numbers) lead to non-isomorphic automorphisms.

The paper establishes a general theorem for non-isomorphism based on tower construction differences.

Abstract

For flows the rank is an invariant by linear change of time. But what we can say about isomorphisms? It seems that in case of mixing flows this problem is the most difficult. However the known technique of joinings provides non-isomorphism for mixing rank-one flows under linear change of time. For automorphisms we consider another problems (with similar solutions). For example, the staircase cutting-and-stacking construction is set by a height $h_1$ of the first tower and a sequence $\{r_j\}$ of cut numbers. Let us consider two similar constructions: one is set by $(h_1, \{r_j\})$, another is set by ($h_1+1, \{r_j\}$), and $r_j=j$. We prove a general theorem implying the non-isomorphism of these constructions.