Symbolic Substitutions, Bratteli Diagrams and Operator Algebras

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.