4 - DYNAMICS OF ℤ^d ACTIONS ON MARKOV SUBGROUPS

Summary

Bruce KITCHENS

IBM T. J. Watson Research Center

Yorktown Heights, NY 10598

U.S.A.

A compact topological group with an expansive ℤ^d action of automorphisms can be represented as a Markov subgroup. These notes are an introduction to the study of their dynamical properties. There are two very different cases: one where the group is totally disconnected and the other where the group is connected. Here the concentration is on the disconnected case but some general ideas are mentioned.

Introduction

These lectures contain an introduction to the study of multi-dimensional Markov shifts which have a group structure. They are called Markov subgroups. The alphabet of the Markov subgroup can be any compact group but we will concentrate on the ones with a finite alphabet. Any expansive ℤ^d action of a compact group can be represented as a Markov subgroup on a suitable alphabet. The formulation of the problems considered and the results presented here can be found in the papers [6], [7], [8], [9] and [10] (1987–1993). A comprehensive introduction to this subject can be found in the book Dynamical Systems of Algebraic Origin by Klaus Schmidt [19]. It contains much more than is presented here and thoroughly treats the case of Markov subgroups with a compact connected alphabet.

The lectures are organized as follows. Section 4.2 contains examples of one-dimensional Markov subgroups and then proves a structure theorem (Theorem 4.2.7) for one-dimensional Markov subgroups on a finite alphabet.