In symbolic dynamics, Subshifts of Finite Type (SFTs) are often used as discretized models for orbit phenomena of various dynamical systems. Two-sided SFTs are bi-infinite paths of a directed graph together with the natural bilateral left shift on them, and are therefore amenable to study via combinatorial and matrix-theoretical techniques.

Despite their apparent simplicity, we still do not know whether the conjugacy problem for SFTs is decidable, and very basic examples still remain mysterious. In work of Williams from 1973, conjugacy of SFTs was shown to have an equivalent matrix-theoretical formulation in terms of adjacency matrices, and was conjectured to coincide with eventual conjugacy. This led to the discovery of various invariants that distinguish SFTs up to conjugacy, as well as a counterexample to Williams conjecture in 1999 by Kim and Roush.

Together with early attacks on Williams conjecture, Cuntz and Krieger found a construction of C*-algebras associated to SFTs that recover several invariants of SFTs, and have led to the discovery of new invariants. This makes several classification problems for Cuntz-Krieger C*-algebras especially relevant to the discovery of new (and computable) obstructions to conjugacy of SFTs. 

In this talk, I will discuss various invariants of SFTs, the relationship between them, how to associate algebras to SFTs, and some of the invariants that are captured by these algebras. Finally, I will explain how C*-algebras of SFTs are classified up to stable equivariant homotopy equivalence in terms of the SFTs, leading to a new way of measuring the obstruction to conjugacy of SFTs. Our proof relies on bimodule theory for C*-algebras, as well as a new bicategorical approach for bimodules initiated by Meyer and his students.

*Based on joint work with Boris Bilich (Haifa U.) and Efren Ruiz (Hawai'i at Hilo).