Symbolic Substitutions, Bratteli Diagrams and Operator Algebras

TL;DR

This paper explores the connections between symbolic substitutions, Bratteli diagrams, and operator algebras, analyzing how various equivalence relations affect their properties and applying these concepts to Fibonacci-like substitutions.

Contribution

It introduces new equivalence relations on substitutions via Bratteli diagrams and examines their impact on properties like pure aperiodicity and primitivity.

Findings

Telescope equivalence preserves pure aperiodicity and primitivity.

Finer equivalence relations do not preserve all invariants.

Application to Fibonacci-like substitutions demonstrates theoretical concepts.

Abstract

In this paper we look at symbolic substitutions and their relationship to Bratteli diagrams and their associated operator algebras. In particular, we consider the equivalence relation on substitutions induced by telescope equivalence of Bratteli diagrams. Such an equivalence preserves pure aperiodicity and primitivity but fails to preserve rank, order, and number of letters. In a similar manner, we consider the equivalence relation on substitutions induced by telescope equivalence of ordered Bratteli diagrams. This results in a finer equivalence but fails to provide a complete invariant. An application to Fibonacci-like substitutions is developed.