Title: Symbolic dynamics through the lens of C*-algebras



Abstract: "In symbolic dynamics, Subshifts of Finite Type (SFTs) are often used as discretized models for various dynamical systems. Two-sided SFTs are bi-infinite paths of a directed graph together with a natural bilateral left shift on them, making them particularly 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 conjugacy for very simple examples still remains mysterious. In work of Williams from 1973, conjugacy of SFTs was shown to have an equivalent matrix-theoretic 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, and eventually to 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 standard invariants of SFTs, and has led to the discovery of new invariants. This makes several classification problems for Cuntz-Krieger C*-algebras particularly relevant to the discovery of new (and computable) obstructions to conjugacy of SFTs. 

 

In this talk, I will showcase some old and new invariants of SFTs, the relationship between them, how to associate algebras to SFTs, and how to recover some of these invariants from the algebras. I will explain how we answered a question of Soren Eilers, showing that eventual conjugacy of SFTs coincides with stable graded homotopy equivalence of the associated C*-algebras. Our proof relies on bimodule theory for C*-algebras, as well as on a new (and surprisingly necessary) bicategorical approach for bimodules initiated by Ralf Meyer and his students.

*Based on joint work with Boris Bilich and Efren Ruiz.


Livestream/Recording Link: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=2ec05120-9f8d-4a64-b12b-b23300ac53f3