Presentations of subshifts and their topological conjugacy invariants

Abstract

We introduce the notions of symbolic matrix system and $\lambda$-graph system that are presentations of subshifts. They are generalized notions of symbolic matrix and $\lambda$-graph for sofic subshifts to general subshifts. We then formulate strong shift equivalence and shift equivalence between symbolic matrix systems and show that two subshifts are topologically conjugate if and only if the associated canonical symbolic matrix systems are strong shift equivalent. We construct several kinds of shift equivalence invariants for symbolic matrix systems. They are the dimension groups, the Bowen-Franks groups and the nonzero spectrum that are generalizations of the corresponding notions for nonnegative matrices. The K-groups for symbolic matrix systems are introduced. They are also shift equivalence invariants and stronger than the Bowen-Franks groups but weaker than the dimension triples. These kinds of shift equivalence invariants naturally induce topological conjugacy invariants for subshifts.