2 INTRODUCTION

Markov shifts, a difficult field of research with no indication yet of a sat-

isfactory general theory. This lack of progress is all the more remarkable

when compared with the richness of the theory in one dimension; it is due

to a variety of reasons, the most famous of which is that any reasonably gen-

eral definition of higher dimensional Markov shifts immediately leads to the

Problem that it may be undecidable whether the shift Space is nonempty. Á

second reason is that none of the techniques which have been so successful

for one dimensional Markov shifts, and some of which were described in

the preceding sections, appear to be applicable here. Section 5 is devoted to

elementary examples of such shifts and to the surprising difficulties these ex-

amples present. However, if one makes the (very restrictive) assumption that

the Markov shift carries a group structure, then many of these difficulties can

be resolved, and one has the beginnings of a successful analysis which turns

out to encompass the theory of expansive AE -actions by automorphisms of

compact groups, and which exhibits an intriguing interplay between commu-

tative algebra and dynamics (Sections 6-7).

The attentive reader will have noticed that there were ten lectures, but

that there are only seven sections in these notes. Since seven and ten have no

common factor, I should explain how the lectures were organized: section one

was covered in two lectures, section two in three lectures, sections three, four,

and five in one lecture each, and the remaining two sections were covered in

two lectures after some of the material from section seven had already been

presented earlier by D. Lind in greater detail in a research seminar.

Copies of a preliminary version of these lecture notes had been distributed

to all members of the audience before the beginning of the lecture series, and

this enabled me to be more selective in the material I presented in the talks

and to make occasional references to the notes for unexplained background

or further details.

These notes have benefitted from many conversations with experts both

before and during the Conference. Special thanks are due to W. Parry and C.

Sutherland for reading critically an earlier draft of these notes (to the latter

also for a number of discussions on equivalence relations on odometers and

Markov shifts), to D. Rudolph for explaining to me the tilings described in

Section 5, to D. Berend for informing me about D. Masser's result [Mas],

and to B. Kitchens for helping me to correct a mistake in Example 2.2(4),

and to him and his friends for feeding me seafood in order to build up

my strength for the talks. However, my sincerest thanks go to D. Lind and

S. Tuncel, who organized both the Workshop and the CBMS Conference in

an exemplary manner, and whose warm hospitality made my stay in Seattle

greatly enjoyable.