Preface

These notes represent a mildly expanded version of a series of ten lectures

that I gave at a CBMS conference organized by Kamel Haddad at California State

University, Bakersfield, CA in June, 1995. Due to external circumstances their

publication has been delayed for a few years but I hope that they still give a

timely presentation of a novel point of view in dynamical systems. I have not

made any effort to update the exposition, but I would like to point out some recent

developments that are relevant to the reader who wishes to pursue matters further.

First of all I would like to recommend the new book by Paul Shields, The ergodic

theory of discrete sample paths, Grad. Studies in Math. v. 13 (AMS), 1996. In

this book there is a very careful treatment of some central issues in ergodic theory

and information from a point of view that is close to that expounded in these

lectures. There is a particularly good treatment there of entropy related matters

and of various characterizations of Bernoulli processes.

Following up on an idea proposed by M. Gromov, Elon Lindenstrauss and I have

developed a new invariant in topological dynamics which refines the classical no-

tion of topological entropy. This invariant, called the mean topological dimension,

vanishes for all systems with finite topological entropy but distinguishes between

various systems with infinite topological entropy. There is a single orbit interpreta-

tion of this invariant which should shed some new light on spaces of meromorphic

functions, solutions of dynamical systems with infinitely many degrees of freedom

etc. The basic theory is set out in a joint paper Mean Topological Dimension, (to

appear in the Israel J. of Math).

The style of these notes is that of a lecture. When proofs are given they are

meant to be complete, but not every i is dotted nor every t crossed. Most of the

material appears elsewhere and I have given references at the end of each chapter

to guide the reader who wants to pursue matters in more detail. Chapter 4 contains

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