On Disjointness of Mixing Rank One Actions

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.