Leningrad Math. J. Vol. 2 (1991), No. 2

On the work of V. A. Rokhlin in ergodic theory

by A. M. Vershik

The scientific heritage of Vladimir Abramovich Rokhlin consists of about sixty publications and several unfinished manuscripts from the 1950's and 60's, and can be conditionally split into four parts: topology - mainly four-dimensional topology and the algebraic apparatus of topology; real algebraic geometry (the last years); ergodic theory; and, finally, work on the history, teaching methods, and methodology of mathematics. The last group, unfortunately, was not reflected sufficiently in his publications, though it occupied Rokhlin constantly, and his ideas, presented often in reports and public lectures, are widely enough known and have had an influence on those around him.

Here we touch briefly on his work in ergodic theory; a detailed analysis of his work in topology and real algebraic geometry is contained in the paper of Viro and Kharlamov. Twenty-four publications were devoted to ergodic theory, along with several unpublished manuscripts which were not finished, unfortunately, and are among his unrealized plans. He returned more than once to the idea of writing a book on the metric theory of dynamical systems, including a large section on general measure theory in his own spirit; as far back as the 1940s he wrote several chapters and proposed continuing the work later with young coauthors, but after ceasing to study ergodic theory, he cooled somewhat to the idea. It should be remarked that some traces of his plans were realized in surveys and books that came out later.

In ergodic theory Vladimir Abramovich introduced a geometric and algebraic culture that it lacked by origin, which was rather analytic in the spirit of the traditions of the theory of dynamical systems of Poincare and, on the other hand, Boltzmann. In this he was continuing the tradition of von Neumann. In Rokhlin's work the geometric side of things (partitions, dynamics, etc.) pre-dominated over the analytic aspects. He proposed that systems of algebraic, analytic, probabilistic, number-theoretic, and other origins should be considered simultaneously. This tradition became established and yielded excellent results in his work and later in the work of his students. On the other hand, he strictly followed axiomatic constructions of measure theory, also affected by the influence of Kolmogorov and von Neumann.

Lebesgue spaces, introduced by Rokhlin in his undergraduate work and thoroughly investigated in the subsequent dissertation and paper ``0n the fundamental concepts of measure theory'', have turned out to be a very successful concept, and his axiomatics an exceptionally convenient refinement of the axiomatizations used previously. It can be said without doubt that, after the axiomatics of Kolmogorov and von Neumann, Rokhlin made the most important step for distinguishing the proper category of measure spaces. Unfortunately, the convenience and importance of the theory of Lebesgue spaces were not realized for some time. It was perhaps for this reason that investigators resorted for a long time to topological concepts for constructions that are in essence purely metric. At present there is no doubt that the category of Lebesgue spaces is a fundamental construction in ergodic theory, measure theory, and other theories. In passing it should be mentioned that Rokhlin also made the important observation that a system of conditional measures, or, as he said, a canonical system of measures, exists only for measurable partitions of Lebesgue spaces, and attempts to introduce them in other categories were incorrect.

If one now asks any specialist in ergodic theory what the two most fundamental results at the basis of the theory are, the answer will be:

  1. the Birkhoff-von Neumann ergodic theorem;
  2. the Rokhlin-Halmos lemma.
This lemma, which is the starting point of all approximation constructions, was proved by Rokhlin at the end of the 1940s, and independently by Halmos in a weaker formulation. It states that, for every aperiodic transformation $T$ of a Lebesgue space $X$ with a finite invariant measure $\mu$, every integer $n$, and every positive number $\varepsilon$, there exists a periodic transformation $T_n$ with period $n$ such that $$ \mu\{x\in X\mid T_nx\ne Tx\}<\frac1n+\varepsilon. $$ This lemma on periodic approximation has been generalized many times; it is the basis for category theorems, approximation theorems, entropy theorems, and so on.

Further, in early work on decomposition of automorphisms into ergodic components, Rokhlin actually proved a variant of a measurable selection theorem now called the Rokhlin-Kuratowski-Ryll-Nardzewski theorem. Influenced by the Gel'fand-Naimark-Raikov-Shilov theory of normed rings, Rokhlin made an (incomplete) attempt to construct a theory of so-called unitary rings. This theory is dual to the theory of Lebesgue spaces, which is a function-analytic version of it.

The first period of development of ergodic theory (the 1930's and 40's) concerned, for the most part, spectral theory. Here Vladimir Abramovich was the author of a number of the results included in his surveys of 1949 and 1958. Widely known are his categorical mixing estimates, the first investigations of automorphisms of compact Abelian groups as dynamical systems, and work on measurable flows.

Many papers, ideas, and initiatives of Vladimir Abramovich were completed or developed in investigations of his students and subsequent authors. These include, for example, the proposal to develop a trajectory theory (R. E. Belinskaya, A. M. Vershik), realization theorems and ``systems over systems'' (we would now say quantization dynamical systems), Gaussian systems (Vershik), the mixed spectrum, fiber bundles (L. M. Abramov), mixing (Ya. G. Sinai), and others.

We should dwell especially on his favorite topic of study in later years (and most important in the 1960's and 7O's) - entropy theory. Kolmogorov's discovery of entropy made a strong impression on Rokhlin. The language of this theory was the language of measurable partitions worked out earlier by Rokhlin and used by Kolmogorov in his work on entropy.

The precise analysis of the concept of the entropy of dynamical systems carried out in a cycle of papers by Kolmogorov and Sinai and then by Rokhlin, Abramov, Pinsker, and others, was simply not possible without the theory of measurable partitions, especially the part of it relating to decreasing sequences. A small and insignificant error in Kolmogorov's first paper, which was discovered by Vladimir Abramovich and made it necessary to give a somewhat different definition of the entropy in the Sinai sense, involved certain subtleties of the theory (see the remark in Kolmogorov's second paper). The unity of the two definitions was finally reestablished considerably later after Rokhlin proved a theorem on generators for aperiodic automorphisms. The concluding results on generators are due here to Krieger.

Vladimir Abramovich's two survey papers in Uspekhi Matematicheskikh Nauk (1960 and 1967) played an enormous role in the development of entropy theory in our country and abroad. The second paper summarizes the development of the concept of entropy and its applications to the theory of transformations with invariant measure.

The theory of invariant partitions for automorphisms especially interested Rokhlin, and he returned to it in later years, in the period when he finished his work on ergodic theory (this was the topic of several unpublished sketches). The start of this theory was the classical joint work of Rokhlin and Sinai in which it was proved, in particular, that the class of $K$-automorphisms coincides with the class of automorphisms with completely positive entropy (this was proved earlier in one direction by M. S. Pinsker). The very first formulas in entropy theory were the Abramov formulas for the derived automorphism and flow; formulas for the entropy of automorphisms of compact groups (Sinai, Arov, Yuzvinskii, and others) were a development of ideas and suggestions of Vladimir Abramovich. Also, the metric properties of automorphisms of compact Abelian groups were investigated (Rokhlin, Sinai, Yuzvinskii) on his initiative.

Rokhlin's interest in ergodic theory gradually began to fall, after the appearance of new post-entropy ideas - approximation and, especially, the fireworks of Ornstein's contributions and that of his successors. Vladimir Abramovich was undoubtedly interested in the course of events, but he did not take part, all the more so because his algebro-topological interests had prevailed by this time.

We mention two more circumstances. For a long time Rokhlin has been interested in number theory and the possibility of applying ergodic theory to it. Although he had only one paper on this topic, and that devoted mainly to the theory of exact endomorphisms (namely, the paper on continued fractions and the Gauss endomorphism), he thought (and this opinion had an indirect confirmation, for example, in the work of Linnik on the ergodic method in number theory) that the possibilities of metric theory in number theory were far from exhausted. Incidentally, Vladimir Abramovich always had an intense interest in the theory of endomorphisms (or semigroups of endomorphisms), and the paper mentioned is a vault of metric concepts and theorems relating precisely to this case. Especially important is the concept of a natural extension; this concept provides an invariant formulation of the immersion of a one-sided process in a two-sided process (metric dilatation). The other circumstance had to do with the interrelations with smooth and classical dynamics. It may seem strange that he, an outstanding specialist in the area of smooth manifolds who had a good knowledge of classical dynamics and physics, did not try to connect ergodic theory with smooth dynamics, all the more so because many of his students, and those who were close to him or felt his influence, studied this topic actively (Sinai, Arnol'd, Anosov, and others). Moreover, communications about the work of Smale, Anosov, and others were heard repeatedly in the seminar. Vladimir Abramovich himself said that here he was an advocate of ``purely'' posed problems not involving a mixture of categories completely unlike each other. In other words, he regarded smooth and metric dynamics as immiscible areas. This point of view was perhaps affected by an echo of axiomatic rigorism, which is now certainly not popular, but one cannot say that it is inconsistent.

At the sources of ergodic theory as a mathematical discipline stand the names of von Neumann and Kolmogorov; after them can be named a few others who shaped this theory from the 1930's to the 1950's and gave it its modern form - G. Birkhoff, S. Ya. Khinchin, E. Hopf, S. Kakutani, P. Halmos, and Vladimir Abramovich Rokhlin.

Work of Vladimir Abramovich Rokhlin on measure theory and ergodic theory

1. 0n classification of measurable partitions, Dokl. Akad. Nauk SSSR 58 (1947), 29 - 32. (Russian)

2. 0n the problem of classification of automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR 58:2 (1947), 189 - 191. (Russian)

3. Unitary rings, Dokl. Akad. Nauk SSSR 59 (1948), 643 - 646. (Russian)

4. A general transformation with invariant measure is not mixing, Dokl. Akad. Nauk SSSR 60 (1948), 349 - 351. (Russian)

5. 0n the fundamental concepts of measure theory, Mat. Sb. 25 (67) (1949), 107 - 150; English transl., Amer. Math. Soc. Transl. (1) 10 (1962).

6. 0n decomposition of a dynamical system into transitive components, Mat. Sb. 25 (67) (1949), 235 - 249. (Russian)

7. On dynamical systems whose irreducible components have purely point spectrum, Dokl. Akad. Nauk SSSR 64 (1949), 167 - 169. (Russian)

8. with A. A. Gurevich, 0n approximation of nonperiodic flows by periodic flows, Dokl. Akad. Nauk SSSR 64 (1949), 619-620. (Russian)

9. 0n endomorphisms of compact Abelian groups, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), 329-340. (Russian)

10. Selected topics in the metric theory of dynamical systems, Uspekhi Mat. Nauk 4 (1949), no. 2 (30), 57 - 123. (Russian)

11. with A. A. Gurevich, Approximation theorems for measurable flows, Izv. Akad. Nauk SSSR, Ser. Mat. 14 (1950), no. 6 (40), 537 - 548. (Russian)

12. Metric classification of measurable functions, Uspekhi Mat. Nauk 12 (1957), no. 2 (74), 169 - 174. (Russian)

13. with S. V. Fomin, Spectral theory of dynamical systems, Proc. Third All-Union Math. Congress (1956), Vol. 3, 1958, p. 284. (Russian)

14. On the entropy of a metric automorphism, Dokl. Akad. Nauk SSSR 124 (1959), 980 - 982. (Russian)

15. New progress in the theory of transformations with invariant measure, Uspekhi Mat. Nauk 15 (1960), no. 4 (94), 3 - 26; English transl. in Russian Math. Surveys 15 (1960).

16. Structure and properties of invariant measurable partitions, Dokl. Akad. Nauk SSSR 141 (1961), 1038-1041; English transl. in Soviet Math. Dokl. 2 (1961).

17. with Ya. G. Sinai, 0n the entropy of an automorphism of a compact Abelian group, Teor. Veroyatnost. i Primenen. 6 (1961), 351 - 352; English transl. in Theory Probab. Appl. 6 (1961).

18. Exact endomorphisms ofa Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 499-5 30; English transl. in Amer. Math. Soc. Transl. (2) 39 (1964).

19. with L. M. Abramov, The entropy of a fiber bundle of transformations with invariant measure, Vestnik Leningrad. Univ. 1962, no. 7 (Ser. Mat. Mekh. Astr. vyp. 2), 5 - 13. (Russian)

20. An axiomatic definition of the entropy of transformations with invariant measure, Dokl. Akad. Nauk SSSR 148 (1963), 779 - 781; English transl. in Soviet Math. Dokl. 4 (1963).

21. Generators in ergodic theory, Vestnik Leningrad. Univ. 1963, no. 1 (Ser. Mat. Mekh. Astr. vyp. 1), 26 - 32. (Russian)

22. Metric properties of endomorphisms of compact Abelian groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 867 - 874; English transl. in Amer. Math. Soc. Transl. (2) 64 (1967).

23. Generators in ergodic theory. II, Vestnik Leningrad. Univ. 1965, no. 13 (Ser. Mat. Mekh. Astr. vyp. 3), 68 - 72. (Russian)

24. Lectures on entropy theory of transformations with invariant measure, Uspekhi Mat. Nauk 22 (1967), no. 5 (137), 4 - 56; English transl. in Russian Math. Surveys 22 (1967)

List of manuscript material of Vladimir Abramovich Rokhlin on ergodic theory

1. Notebook of small format in black binding without heading, about 80 pages, materials for a book (apparently from the 1940's), Table of contents: Part I. Lebesgue spaces, Chapters 1 - 13.

2. Sketch: ``Foreword''. Organization of the book. History of the theory of transformations with invariant measure, its connections and applications. Measure theory as an independent science, and the true place of the theory of transformations with invariant measure. What is usually understood by measure theory; purpose of the first chapter, what is assumed known. Main goal of the book - new things. Begin with this. Characteristics of the old parts. Selection of materials. Degree of generality - Lebesgue spaces.

3. Manuscript, ``Transformations with invariant measure''. Written parenthetically: ``book'', 69 pages + 29 (apparently later) - probably relates to the 1960's.

4. Manuscript, ``Unitary rings'', 18 pages (apparently from the 1950s).

5. Ergodic theory 1966 - 1967. Lectures (plans), 4 pages.

6. Invariant partitions, June 1967, 2 pages; additions - July 1967, September 1967.

7. ``Closed partitions. Report October 1, 1968'', 1 page; additions - October 11 and 21, 1968.

8. ``Saturated partitions. December 1969'', 2 pages, 24 items.

Translated by H. H. MCFADEN