Branch points of substitutions and closing ordered Bratteli diagrams

TL;DR

This paper investigates the conditions under which stationary ordered Bratteli diagrams produce continuous Vershik maps, providing algorithms and representations for substitution subshifts with specific branch point properties.

Contribution

It introduces necessary and sufficient conditions for continuous Vershik maps in stationary ordered Bratteli diagrams and presents algorithms for identifying branch points in substitutions.

Findings

Characterization of when stationary ordered Bratteli diagrams generate continuous Vershik maps.

Algorithms for finding branch points in substitutions.

Adic representations for substitutions with one or fixed branch points.

Abstract

We study stationary ordered Bratteli diagrams and give necessary and sufficient conditions for these orders to generate a continuous Vershik map. We apply this to finding adic representations for one sided substitution subshifts. We give an algorithm to find the branch points of a substitution, which have to be mapped to the minimal elements of such an ordering. We find adic representations for substitutions with one branch point, and also substitutions all of whose branch points are fixed.