A Family of Totally Rank One Two-sided Shift Maps

TL;DR

This paper introduces a family of totally rank one two-sided shift maps, expanding the understanding of rank one transformations through generalized del Junco-Rudolph's maps and related skew products.

Contribution

It demonstrates that all maps within the generalized del Junco-Rudolph's family are totally rank one, providing new insights into their structural properties.

Findings

All generalized del Junco-Rudolph's maps are totally rank one.

A relative prime relation $h_{k+1} ot rsim 1 mod q$ is verified.

The structure of skew products is linked to the rank one property.

Abstract

'Generalized del Junco-Rudolph's map', a sub-family of generalized Chacon's map (\cite{FER1}), is introduced. A skew product related to the structure of the Generalized del Junco-Rudolph's map is introduced. A Relative Prime Relation, $h_{k+1}\equiv 1 \mod q$ is verified, based on the proposition of iterations of this skew product. We say a measure-preserving transformation $T$ is totally rank one if $T^{q}$ is rank one for every $q\in \mathbb{N}$. In this paper, we show that of every Generalized del Junco-Rudolph's map is totally rank one.