Chacon's Type Ergodic Transformations with Unbounded Arithmetic Spacers

TL;DR

This paper explores generalizations of the Chacon map with unbounded spacer sequences, revealing diverse ergodic properties such as rigidity, minimal self-joinings, and mixing behaviors, and discusses open questions in this class of transformations.

Contribution

It introduces new classes of rank-one transformations with unbounded spacers and analyzes their ergodic properties, expanding the understanding of Chacon-type systems.

Findings

Root sequences like s_j= [√j] are rigid and have all polynomials in their weak closure.

Linear spacer sequences s_j= j exhibit minimal self-joinings (MSJ).

Transformations show partial rigidity and various mixing properties.

Abstract

The following generalizations of the Chacon map are proposed: instead of classical constant spacer sequence $(0,1,0)$ let a sequence $(0,s_j,0)$ be one with unbounded $s_j$. (We mention also an analogue of the historical Chacon map with spacer sequences in the form $(0,s_j)$.) This narrow class of rank-one transformations may be abundant source of open questions. All such constructions have partial rigidity, but some other properties could be different. For root sequence, $ s_j= [\sqrt{j}]$, (or $ s_j= [\ln{j}]$) the corresponding action is rigid, moreover it possesses all polynomials in its weak closure. In the linear case $s_j={j}$ we get (as well as for the classical Chacon transformation) the property of minimal self-joinings (MSJ). We present some observations about MSJ, mild mixing, partial mixing, $\ae$-mixing, absence of factors, triviality of centralizer and spectral primality, state several problems, and mention exponential "self-similar" Chacon transformations and flows on infinite measure spaces.