Limited scope adic transformations

TL;DR

This paper introduces a new family of nonstationary, nonsimple adic transformations on diagrams, analyzes their dimension groups, and establishes ergodic properties, including the loosely Bernoulli nature of certain transformations like the Euler adic.

Contribution

It defines a new class of adic transformations, computes their dimension groups, and characterizes their ergodic measures, extending previous studies to more general nonstationary cases.

Findings

Explicit dimension groups for a subfamily of adic transformations

Identification of ergodic invariant measures for the subfamily

Demonstration that the Euler adic is loosely Bernoulli

Abstract

We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations. We relate the dimension group of each these diagrams to the dynamical system determined by the adic transformation on the infinite edge paths, and we explicitly compute the dimension group for a subfamily. We also determine the ergodic adic invariant probability measures for this subfamily, and show that each system of the subfamily is loosely Bernoulli. We also give examples of particular adic transformations with roots of unity as well as one which is totally ergodic called the Euler adic. We also show that the Euler adic is loosely Bernoulli.