Rokhlin's School in ergodic theory

Abstract

Ergodic theory is one of several fields of mathematics where the name Vladimir Abramovich Rokhlin (spelled also 'Rochlin' and 'Rohlin') is well known to the specialists. That name is attached to some very often used theorems, but the goal of this paper is not just to remind the reader of these theorems. I put them in the context of the general development of ergodic theory during the thirty years 1940-70. Most of all, I want to emphasize that Rokhlin was not only a researcher producing powerful results but also a founder of schools at the two best Universities in the USSR. For at least 10 years (1958-68) these schools dominated ergodic theory. This paper is not biographical. Rokhlin's life certainly deserves a better biographer. However, I mention certain circumstances of a non-mathematical nature where it seems to be appropriate. This paper is divided into three sections corresponding to three easily distinguishable periods in Rokhlin's work in ergodic theory. The first section called BE (before entropy) corresponds to the first period which lasted from 1940 to 1949. It was devoted to axiomatization, spectral invariants, and mixing properties. The following 9 years were a quiet period in ergodic theory. In particular, Rokhlin was working in topology and apparently did not plan to return to ergodic theory. In 1958, A. Kolmogorov introduced a new invariant: the entropy of a measure preserving transformation. This started a new era in ergodic theory that I will call ET (entropy theory) and attracted Rokhlin back to the area. The second section of this paper is devoted to the results of the seminar organized by Rokhlin at the Moscow State University and covers approximately 1958-1959. We also follow the later history of several important problems formulated in this seminar. Finally, the third section is devoted to Rokhlin's seminar at the Leningrad State University in 1960 on the isomorphism of Bernoulli automorphisms with equal entropy started yet another period in ergodic theory. This period can be characterized by the development of methods of more combinatorial nature that were harder to formalize. Many old problems were solved by these methods. At this point the center of pure ergodic theory or rather Bernoulli theory moved to the West. Perhaps this was one of the reasons why Rokhlin stopped doing active research in the area. Besides, he was busy proving his famous results in 4-dimensional topology.