Fifty years of entropy in dynamics: 1958--2007

Anatole Katok
2007 Journal of Modern Dynamics  
PREFACE These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more "historical" nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fifty-year period under discussion and on numerous occasions
more » ... s fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre. These notes are meant as "easy reading" for a mathematically sophisticated reader familiar with most of the concepts which appear in the text. I recommend the book [59] as a source of background reading and the survey [44] (both written jointly with Boris Hasselblatt) as a reference source for virtually all necessary definitions and (hopefully) illuminating discussions. The origin of these notes lies in my talk at the dynamical systems branch of the huge conference held in Moscow in June of 2003 on the occasion of Kolmogorov's one hundredth anniversary. The title of the talk was "The first half century of entropy: the most glorious number in dynamics". At that time not quite half a century had elapsed after the Kolmogorov discovery, although one could arguably include some of the "pre-history" into the counting. Now this ambiguity has disappeared, and I dedicate this article to the fiftieth anniversary of the discovery of entropy in ergodic theory by Andrei Nikolaevich Kolmogorov. There is a number of published recollections and historical notes related in various ways to our subject. I used some information from these sources with references, although this plays a secondary role to personal recollections and analysis based on the knowledge of the subject and its historical development. Naturally I took all necessary care not to contradict any source of this kind. I do recommend the account by M. Smorodinsky [140] which addresses some of the
doi:10.3934/jmd.2007.1.545 fatcat:6muz4yjqdvas3c7thlvqgv4h6y