Shift equivalence through the lens of C*-correspondences.
Adam Dor-On (Haifa University)
Thursday 18th April 16:00-17:00 Maths 116 (Zoom Talk)
Abstract
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.
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 Gottingen) and Efren Ruiz (Hawai'i at Hilo)
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