Characters on the Full Group of the Odometer

TL;DR

This paper introduces a dense subgroup of the full group of the odometer transformation and characterizes all indecomposable characters on this subgroup and the full group, advancing understanding of their algebraic structure.

Contribution

It defines a new dense subgroup of the full group of the odometer and provides a complete description of its indecomposable characters, extending to the full group.

Findings

Characterization of indecomposable characters on the subgroup

Extension of character description to the full group

New structural insights into the full group of the odometer

Abstract

Let $X$ be the space of all infinite $0,1$-sequences and $T$ be the odometer on $X$. In this paper we introduce a dense subgroup $S(2^\infty)$ of the full group $[T]$ and describe all indecomposable characters on $S(2^\infty)$. As result we obtain a description of indecomposable characters on $[T]$.