3 - ALGEBRAIC ASPECTS OF SYMBOLIC DYNAMICS

Summary

Mike BOYLE

Department of Mathematics

University of Maryland

College Park, MD 20742–4015 U.S.A.

This is an introduction to some algebraic aspects of symbolic dynamics, with emphasis on shifts of finite type, based on four lectures at the Temuco School.

Introduction

These four lectures are an introduction to some algebraic aspects of symbolic dynamics, with no assumption of previous background. The intent is much more to describe ideas and invariants than to give proofs, and this tendency becomes more pronounced in the last two lectures.

Lecture I (sections 3.2 - 3.6) is elementary background on subshifts. The sections 3.2 - 3.3 are elementary (but cover a lot of ground) and the proofs left to the reader are feasible.

Lecture II (section 3.7) describes and mostly explains the standard matrix invariants for shifts of finite type.

Lecture III (section 3.8) explains the basics of dimension groups and shift equivalence, with some hints at further developments. There is a greater emphasis on the order structure of dimension groups than we need for the application in this lecture. This reflects the fundamental importance of the order structure of these groups in other situations.

In Lecture IV (sections 3.9 – 3.11), we throw proof to the winds to describe one of the deepest results on SFT's, the Kim-Roush-Wagoner Factorization Theorem, and the Kim-Roush counterexamples to Williams' Conjecture.