Bratteli-Vershik models for partial actions of $\mathbb{Z}$

TL;DR

This paper demonstrates that partial actions of the integers on the Cantor set, induced by a homeomorphism with dense orbits, can be represented as Vershik maps on Bratteli diagrams, with all such diagrams being equivalent.

Contribution

It extends the Bratteli-Vershik model to partial actions of a7, showing their realization and equivalence through Bratteli diagrams.

Findings

Partial actions can be realized as Vershik maps.

Any two Bratteli diagrams for such actions are equivalent.

The approach generalizes existing models for full actions.

Abstract

Let $U$ and $V$ be open subsets of the Cantor set with finite disjoint complements, and let $h:U\to V$ be a homeomorphism with dense orbits. Building from the ideas of Herman, Putnam, and Skau, we show that the partial action induced by $h$ can be realized as the Vershik map on a Bratteli diagram, and that any two such diagrams are equivalent.