Symbolic dynamics and the stable algebra of matrices

TL;DR

This paper explores the stable algebra of matrices in symbolic dynamics, highlighting its algebraic properties, connections to K-theory, and implications for shift equivalence and automorphism groups.

Contribution

It introduces the stable algebra of matrices over various rings, linking symbolic dynamics with algebraic K-theory and providing counterexamples to longstanding conjectures.

Findings

Connection between stable algebra and algebraic K-theory

Counterexamples to Williams' Shift Equivalence Conjecture

Analysis of automorphism groups of shifts of finite type

Abstract

We give an introduction to the "stable algebra of matrices" as related to certain problems in symbolic dynamics. We consider this stable algebra (especially, shift equivalence and strong shift equivalence) for matrices over general rings as well as various specific rings. This algebra is of independent interest and can be followed with little attention to the symbolic dynamics. We include strong connectionsto algebraic K-theory and the inverse spectral problem for nonnegative matrices. We also review key features of the automorphism group of a shift of finite type, and the work of Kim, Roush and Wagoner giving counterexamples to Williams' Shift Equivalence Conjecture.