Abstract:

To each directed graph, one can associate a dynamical system with discrete time, consisting of bi-infinite paths, where the evolution mapping is defined by the path shift. Such a dynamical system is called a shift of finite type and it is a central object of study in symbolic dynamics.

One of the main open questions in this area is classification of shifts of finite type up to conjugacy and eventual conjugacy. In a foundational work from 1973, Williams reduced this problem to the classification of incidence matrices of corresponding graphs up to shift (SE) and strong shift (SSE) equivalence. Williams presented a reasonable classification of matrices up to shift equivalence and hypothesized that SE and SSE coincide. Nearly 20 years later, this was refuted by Kim and Roush through a counterexample.

This classification problem is closely related to C*-algebras of graphs. It turns out that two graphs with SSE incidence matrices have stably isomorphic C*-algebras. Furthermore, if we equip the C*-algebras of graphs with two additional structures: a commutative diagonal subalgebra and a gauge action of the circle, then we obtain a complete invariant of strong shift equivalence. In our work with Dor-On and Ruiz, we show that stable equivariant homotopic equivalence of C*-algebras is equivalent to shift equivalence of graphs.

In the talk, I will discuss these constructions and results in more detail and explain how the perspective of C*-algebras can help resolve open questions in symbolic dynamics.