The history of V. A. Rokhlin's ergodic seminar (1960--1970)
aa r X i v : . [ m a t h . HO ] F e b The history of V. A. Rokhlin’s ergodic seminar(1960–1970)
A. M. Vershik ∗ Abstract
The paper tells about the main features and events of the ergodicseminar organized and headed by V. A. Rokhlin at the Leningrad StateUniversity. The seminar was active in 1960–1970.
1. V. A. Rokhlin’s moving to Leningrad
Vladimir Abramovich Rokhlin, with his wife Anna Alexandrovna Gurevich,his son Volodya (born in 1952), and his daughter Lisa (born in 1955), movedto Leningrad in the late summer of 1960. Before, they lived in Kolomnanear Moscow, where V.A. worked at a pedagogical institute. In 1959, therector of the Leningrad State University A. D. Alexandrov offered V.A.a position of professor at the Chair of Geometry, of which he was the head.The corresponding advice Alexandrov received from a friend and colleagueN. V. Efimov, a well-known Moscow geometer, whose family were old friendsof the Rokhlins: Efimov’s wife was a friend of A.A. from the 1930s, whenthey both were L. S. Pontryagin’s graduate students in Voronezh.Immediately upon arrival in Leningrad, in September 1960, V.A. an-nounced his research seminar on the theory of dynamical systems (“ergodicseminar”). Besides, after a while he started to teach an elective course incombinatorial topology. Neither the first area of mathematics (the theory ofdynamical systems), nor the second one (modern topology) was representedat the university before. Another seminar, on topology, was announcedby V.A. somewhat later, since it was necessary first to train at least thefirst group of participants. This seminar was started in 1961, became wellknown, and existed almost until V.A.’s death in 1984. It deserves a separatememoir. ∗ St.Petersburg Department of Steklov Institute of Mathematics, St.Petersburg, Russia,email: [email protected] . The prehistory Here I want to present the history of the ergodic seminar. Its rapid creationand immediate activity are explained, first, by the fact that it coincidedwith an impressing progress in the theory of dynamical systems caused bythe appearance of Kolmogorov’s entropy and other events, succeeded by anenormous interest to the subject. But there was also another circumstance:the ground for the seminar was well prepared. V.A.’s arrival in Leningradwas preceded by one year by the arrival of his first graduate student indynamical systems Leonid Mikhaylovich Abramov, whose wife was fromLeningrad. In 1955, Abramov graduated from the Dnepropetrovsk Univer-sity with a degree in approximation theory and was sent to work at theArkhangelsk Pedagogical Institute, where V.A. held a position from 1952,when he was forced to leave Moscow after he had been effectively expelledfrom the Steklov Institute of Mathematics by its director I. M. Vinogradov.L. M. Abramov, on V.A.’s advice, began to study ergodic theory, andin the mid-1950s enrolled in the postgraduate program of the Moscow StateUniversity under V.A.’s supervision (by agreement with A. N. Kolmogorov).L.M.’s thesis became the first PhD thesis on entropy theory (defended atthe Moscow State University in 1960); he himself was the first V.A.’s grad-uate student. Many knew about V.A.’s plans of moving to Leningrad, some(me included) knew from Moscow friends about L.M.’s arrival. Several peo-ple, including O. A. Ladyzhenskaya, who was friends with V.A. from thetime of her university studies in Moscow, helped L.M. to solve the prob-lem, extremely difficult at the time, of getting a job at the Leningrad StateUniversity. Coincidentally, at this very time L. V. Kantorovich organizeda new department of mathematical economics at the Faculty of Economics,and this made it possible to get L.M. a job at the Chair of MathematicalEconomics, where he worked till the end of his life. The times were like that.In Leningrad, at G. P. Akilov’s (1921–1986) seminar, there was inter-est in measure theory on locally compact groups and especially in mea-sure theory on linear topological spaces (the influence of I. M. Gelfand andD. A. Raikov). The core members of the seminar were B. M. Makarov,V. P. Khavin, V. N. Sudakov, and myself. Some of us had already heardabout Kolmogorov’s entropy. In the fall of 1956, the Second All-UnionConference on Functional Analysis was held in Odessa. Like the first one(which took place in Moscow in January 1956), it attracted many well-knownspecialists. In particular, S. V. Fomin gave a talk about the famous Kol-mogorov’s paper on entropy (
Dokl. Akad. Nauk SSSR , 1958), which had justappeared. At the next, third, conference on functional analysis, which was2eld in Baku in the spring of 1960, V.A. gave a talk about the progress in thetheory of dynamical systems related to the notion of entropy. Together withother graduate students from Leningrad (B. M. Makarov, V. N. Sudakov),I participated in both conferences. We were very impressed by these talks,and they aroused a great interest in the subject.An interesting episode occurred at the Odessa conference. After S. V. Fo-min’s talk about Kolmogorov’s entropy, an unknown graduate student ofthe Odessa University D. Z. Arov unexpectedly rose from his seat to claimthat A. N. Kolmogorov’s results, presented in the talk, on applications ofShannon’s entropy to Bernoulli schemes were contained in his master thesis,of which he informed A.N. in a letter. A.N.’s second paper on entropy(
Dokl. Akad. Nauk SSSR , 1959) does indeed mention D. Z. Arov’s letter.But what exactly had been done by D. Z. Arov and how this is related tothe Kolmogorov–Sinai entropy became clear only quite recently. The partof his master thesis related to information theory was recently publishedin our series of
Zapiski Nauchnykh Seminarov POMI (Vol. 436, 2015) withcommentaries written, at my request, by B. M. Gurevich. Thus, D. Z. Arovmay be safely called one of the pioneers of entropy ergodic theory.Later, in 1962, D. Z. Arov came to Leningrad in order to work with V.A.and took an active part in the ergodic seminar, but then he switched to afield close to his teacher M. G. Krein.In the 1959–1960 academic year, we organized a home student seminar onmeasure theory and functional analysis, where we studied books by A. Weil,P. Halmos, and some papers on integration theory and measure theory.It is appropriate to recall V.A.’s visit to Leningrad in 1957 to give a talkat G. M. Fikhtengolts’s and L. V. Kantorovich’s seminar. I was present atthis meeting and remember the talk: it was one of the last meetings of theseminar, soon it ceased to exist because of L.V.’s moving to Novosibirsk.The talk was devoted to the recent V.A.’s paper in
Uspekhi Mat. Nauk onmetric invariants of measurable functions based on his theory of measurablepartitions. I remembered this talk, and many years later used and general-ized it in several directions. During the same visit (or another one close intime), V.A. gave a lecture at the Leningrad Department of Steklov Instituteof Mathematics on recent progress in algebraic and combinatorial topology.A rotaprint edition of the text of the lecture was immediately issued, andfor a long time it attracted the attention of young mathematicians.3 . The early days of the seminar and its partici-pants
Upon arrival, V.A. asked L. M. Abramov, who by September 1960 alreadyhad good contacts with Leningrad mathematicians, to put him in touch withthose wishing to take part in an ergodic seminar. The first organizationalmeeting, as far as I remember, took place not later than October 1960 atthe Faculty of Mathematics and Mechanics. L.M. became the secretary ofthe seminar.The fact that V.A. was now a professor at the Faculty of Mathemat-ics and Mechanics was known to many; the interest to him and to thenew seminar was great. That is why, the first meeting attracted manyundergraduates, graduate students, and professors. V.A. proposed to be-gin with the study of measure theory in his understanding, the theory ofKolmogorov’s entropy, as well as a series of classical works on dynamics:spectral theory, automorphisms of compact groups, geodesic flows, relationto stationary random processes, etc. All these subjects were a completenovelty for most of the audience. Now I remember only those who tookpart in the seminar for a sufficiently long time in its early years. Amongthem, I name only a few: L. M. Abramov, I. A. Ibragimov, V. N. Sudakov,B. M. Makarov, A. M. Kagan, R. A. Zaidman. Later, during the 1960s,some young mathematicians, who had graduated from the University orthe postgraduate program, joined in: R. M. Belinskaya, S. A. Yuzvinsky,M. I. Gordin, A. A. Lodkin, A. N. Livshits, and others. Of course, therewere those who had just flickered and disappeared, there were, especially atfirst, curiousity seekers; the corresponding list is long and difficult to recover.The first meeting of the seminar was opened by V.A., who told about hisplans for the seminar and suggested the topics of the first talks (see below).These topics were rather quickly taken up, and soon regular meetings began.The first invited speakers included two leading Leningrad mathematicianswho worked, in particular, on problems close to dynamics and with whomV.A. maintained close contact: D. K. Faddeev, who spoke about the classi-fication of automorphisms of lattices, and Yu. V. Linnik, who gave a generaloutline of his recent paper on the equidistribution of integer points on man-ifolds. I have kept a copy of the first paper (later transformed into the book
Ergodic Method and L -Functions ), which was presented by Yu.V. to V.A.and passed to me by V.A.’s widow, bearing the inscription: “This is my bestpaper.”An important feature of the seminar was the abundance of guests, mainly4rom Moscow, where Ya. G. Sinai’s and V. M. Alekseev’s seminar on dynam-ical systems was very active at the time. This seminar followed A. N. Kol-mogorov’s initiative of the late 1950s to revive the theory of dynamicalsystems in Moscow and, in particular, to assert entropy theory, which wasemerging after his pioneering works, as one of its topics. At A.N.’s proba-bility seminar, this topic received some (little) attention, and V.A., whichat the time worked near Moscow, took part in it. He immediately appre-ciated A.N.’s idea and became one of the main developers of this theory.Recall that in his second note on entropy, A.N. emphasizes the role of thetheory of measurable partitions created by V.A. and in a footnote cites thecounterexample suggested by V.A. to one of the statements of the first note.V.A. repeatedly mentioned the very great impression which Kolmogorov’swork on entropy made on him. To this work he attributed his own renewedinterest to the theory of dynamical systems, the subject of his famous pa-pers of the late 1940s, which made him (along with the subsequent workson entropy theory) a leading expert in this field.All of these things largely predetermined also the topics of the Leningradergodic seminar. And they explain the fact that its guests and speak-ers included all active Moscow “dynamists” of the 1960s: V. I. Arnold,Ya. G. Sinai, D. V. Anosov, V. M. Alekseev, and somewhat later their pupilsB. M. Gurevich, V. I. Oseledets, A. B. Katok, A. M. Stepin, G. A. Margulis,and others. There were also guests from other cities: Odessa, Tashkent,Tbilisi, Novosibirsk, Gorky, etc.The seminar existed up to the late 1960s. Occasional meetings (say, onthe occasion of visits or special events) took place even later. I rememberthat I spoke about my Doctor of Sciences dissertation in 1971. In 1965, Istarted my own seminar on smooth dynamics, whose participants includedsome of those I have already mentioned, and also M. L. Gromov, V. L. Eidlin,S. M. Belinsky, and others. The seminar existed for one and a half year.In 1968, I organized a wider seminar on measure theory in linear spaces,operator algebras, representation theory, and dynamics, which exists to thisday.
4. Topics of the seminar
The main topic of the ergodic seminar was, of course, ergodic theory ratherthan the entire dynamics. V.A. emphasized that he advocated distinguishingbetween mathematical structures and preferred not to mix, say, smooth dy-namics with metric or topological dynamics. At the same time, he knew clas-5ical dynamics (both smooth and number-theoretic) well, but for him theyexisted separately, and he gave precedence to metric (measure-theoretic) dy-namics; I dare say, for aesthetic reasons. Moreover, he believed, for example,that it is the methods of the theory of measure-preserving transformationsthat would eventually allow one to solve many problems of both analyticnumber theory and smooth dynamics.He particularly emphasized the role of dynamical systems of algebraicorigin as natural examples. Topological dynamics, as compared with metricdynamics, was for him a less natural area of mathematics. More exactly,he put emphasis on the existence of an invariant measure with respect toa group of transformations. Of course, such a measure does not alwaysexist, but then this is rather a flaw in the problem formulation, and it isthe invariance of a measure with respect to some group of transformationsthat ensures the meaningfulness of studying this measure. On the otherhand, V.A. believed that measure theory itself is not an especially impor-tant field, in particular, thought that the theory of nonseparable measurespaces is a nonessential and somewhat pathological object. He believed thatLebesgue spaces, introduced and studied by him, cover the needs of analysisand dynamics in measure spaces. The widespread clich´e “measure space”without further details invited his criticism. He preferred the more accu-rate term “the theory of transformations with an invariant measure” to thevague term “ergodic theory.” I plan to express my views on the role of V.A.’sseminal paper “On the fundamental ideas of measure theory” and on themodern understanding of what is the category of measure spaces elsewhere.Concluding the general description of the seminar, I want to list severalprinciples which, as I think, guided V.A. in mathematics in general and,in particular, in his style of conducting a research mathematical seminar.First of all, he insisted that all statements must be clear-cut and, if possible,structurally (or categorically) pure. Further, he insisted on the invariance(functoriality) of statements and, again if possible, proofs. He rejected ar-guments that went beyond the framework of a given structure or category,and this occasionally caused discussions, which usually ended in his favor.This attitude could be regarded as an extreme kind of Bourbakism, butV. I. Arnold, a well-known enemy of Bourbaki, eagerly supported V.A.’srecommendations and followed them.Besides, V.A. was able to appreciate the beauty of clear-cut statementsand the originality of arguments, and this feature is in a sense opposite toany dogmatism. Anyway, the school of V.A.’s seminar had much to offerto those who went through it and, possibly, did not subscribe to all hisprinciples. 6 . On the style of conducting the seminar and thefirst meetings
Unfortunately, the agenda of all seminar meetings for almost ten years ofits existence has not survived. Now it seems astonishing: thanks to modernelectronic techniques, we easily save anything and everything. But at thetime, it was necessary to keep records, follow the sequence of talks, whichonly few were capable of. As a result, we (and the history of science) havelost very much. I wrote about this in relation to the famous I. M. Gelfand’sseminar (see M. A. Shubin’s notes of talks at the seminar available in theinternet); but this applies as well to other seminars.The above-mentioned numerous guests of the seminar spoke about theirpublished papers; it is hardly worthwhile (and anyway impossible) to listthem or reproduce the corresponding discussions. Of course, V.A.’s opinionon a result or a talk, with which he usually concluded the meeting, as well ashis judgements and comments, were always of interest both to the audienceand speakers. He always expressed these opinions with authority and reserveno matter whether the talk was received with enthusiasm or with doubt. Itis worth mentioning that V.A. disapproved of other styles of conductinga seminar, when the leader, albeit a distinguished scholar, reacts to talkswith unrestrained emotions. The culture of mathematical seminars in Soviettime was being worked out gradually, following, apparently, European andpre-revolutionary traditions. V.A. followed the best of them.It remains to extract from my memory what it has preserved. Unsurpris-ingly, I remember mainly seminars somehow related to subjects of interestto myself, and some of my own talks. To begin with, I present only a partiallist of talks of the first period about published results given mainly by theparticipants of the seminar.During the initial period, in 1960–1961, the talks were about famouspapers on ergodic theory and measure theory: about the Lebesgue space(V. N. Sudakov), the definitions of entropy (I. A. Ibragimov), spectral the-ory (L. M. Abramov, quasidiscrete spectrum; B. M. Makarov, the spec-trum of Gaussian systems), the martingale theorem and ergodic theorem(A. M. Kagan), etc.I will write in slightly more detail about my own talks of the initial pe-riod. First, V.A. asked me to tell about his little-known paper on unitaryrings, which he valued. It contained an attempt, in the spirit of Gelfand–Naimark and Moscow functional analysis, to present measure theory andthe notions of ergodic theory in terms of the space L of square-integrable7unctions. For this, the Hilbert structure of the space of functions had to beequipped with some additional structure, and V.A. choose a structure whichhe called a unitary ring (later, it became known as a unitary algebra). Inother words, one considers an axiomatization of an unbounded (i.e., definednot for all pairs of elements) multiplication in L with obvious constraints.Indeed, this defines a functor between the corresponding categories (the cat-egory of Lebesgue spaces and the category of unitary rings; the terms did notexisted at the time). To an automorphism with an invariant measure therecorresponds a multiplicative unitary operator. This is a natural measure-theoretic counterpart of Gelfand’s theory of normed rings. However, anothercategory has become more popular here, whose objects are L spaces witha cone, the cone of nonnegative functions. This functor proved to be moreconvenient, for example, when dealing with operators: nonnegative auto-morphisms or (later) polymorphisms and Markov operators. But my talkcontained an answer to a specific question indirectly posed by V.A.: howcan one define a structure of a unitary ring on an infinite-dimensional spacewith a Gaussian measure?At the time, V. N. Sudakov and myself were interested mainly in geo-metric aspects of Gaussian measures (quasiinvariant measures, extension ofweak distributions, etc.), partly under the influence of the Moscow school(I. M. Gelfand, D. A. Raikov). In particular, I studied N. Wiener’s book Nonlinear Problems in Random Theory (1958). But the interest to the dy-namics of linear automorphisms of Gaussian measures, or, as we said backthen, “normal dynamical systems,” was initiated by A. N. Kolmogorov andpicked up by V.A., S. V. Fomin, and others. This applied to spectral theory,entropy, etc. A.N.’s remarkable idea, implemented by his pupil I. V. Gir-sanov, led to constructing an automorphism with simple singular spectrum(in the complement to the constants). V.A. popularized this field, thosewho worked in it included L. M. Abramov and later Ya. G. Sinai and manyothers, including myself. In my talk, I described multiplication formulasfor generalized Hermite polynomials which defined the structure of a uni-tary ring. Then, and somewhat later, I understood that these formulas areequivalent to formulas from the Itˆo–Wiener stochastic analysis, or Wick’sregularization formulas (in physics), etc.I remember that V.A. liked these results very much. Later, they wereincluded in my PhD thesis. Back then, I was a graduate student, formallyunder the supervision of G. P. Akilov, a pupil and coauthor of L. V. Kan-torovich, to whom I owe much, but actually my thesis was inspired by V.A.Among other theorems proved in the thesis, it announced a result on theisomorphism of Gaussian automorphisms with absolutely continuous spec-8rum, which was proved only for some special spectra. In full generality itwas proved somewhat later by D. Ornstein. Afterwards, I repeatedly spokeabout this at the seminar.I remember some of my other talks, initiated by V.A., about famouspapers, for example, my talk about the paper “Geodesic flows on manifoldsof constant negative curvature” by I. M. Gelfand and S. V. Fomin (
UspekhiMat. Nauk , Vol. 7, No. 1(47), 1952) and the corresponding discussion, aswell as my talk about S. Smale’s remarkable paper on the orthogonal groupas a retract of the group of homeomorphisms.
6. Talks of seminar participants about their ownresults
Almost all participants of the ergodic seminar worked, apart from ergodictheory, also in other fields of mathematics, and even reported the obtainedresults at the seminar. I remember V. N. Sudakov’s talks about measuresin functional spaces and my own talks about the axiomatics of measures inlinear spaces from the standpoint of Lebesgue spaces, etc. I had alreadybegun to work in representation theory, algebraic and combinatorial asymp-totics, always bearing in mind connections and analogies with dynamics.But, speaking about not educational or “exterior” talks, but research talkson the topic of the seminar, irrespective of the time when they were given,I must first of all mention, at least very briefly, the most important resultsof the 1960s–1970s obtained by the permanent participants of the seminarin ergodic theory proper.1. V.A.: a proof of the existence of a countable generator for everyaperiodic endomorphism; later, he proved the existence of a generator withfinite entropy for endomorphisms with finite entropy. The ultimate result onthe existence of a finite generator for such endomorphisms with an estimatewas proved by W. Krieger and, independently, by my graduate studentA. N. Livshits. But V.A.’s result was the first in this direction.2. V.A. and L. M. Abramov: a formula for the entropy of skew productsusing the new notion of the mixed entropy of fibers.3. V.A. and Ya. G. Sinai: a proof of the fundamental theorem on thecoincidence of the classes of Kolmogorov automorphisms and of automor-phisms with totally positive entropy; this is one of the fundamental factsof ergodic theory. Later, it was extended by D. Ornstein and B. Weiss toactions of amenable groups.The next two permanent participants of the seminar from the mid-9960s deserve special mention. During the years of V.A.’s teaching at theLeningrad State University, he had many PhD and master students in topol-ogy, but, apart from his very first PhD student L. M. Abramov, only twoPhD students in ergodic theory. The first of them was S. A. Yuzvinsky, whobrilliantly completed his evening studies at the Faculty of Mathematics andMechanics of the Leningrad State University (earlier, he had graduated fromthe Polytechnic Institute), but nevertheless, in spite of V.A.’s recommenda-tion, was not admitted to the postgraduate program at the University. Heenrolled in such a program at the Hertzen Institute, where V.A. workedpart-time for a while, and under V.A.’s supervision prepared and defendedhis PhD thesis (in 1966).4. S. A. Yuzvinsky’s results on the entropy theory of automorphismsof compact groups. This was one of V.A.’s favorite topics. He himself andmany his pupils and colleagues (Ya. G. Sinai and his pupils, D. Z. Arov,L. M. Abramov) studied entropy formulas for group automorphisms. S. A.’sresults and subsequent work by K. Schmidt and others (on the Mahler mea-sure) in a sense settled this problem. Another specific result obtained byS. A., on the genericity of automorphisms with simple continuous spectrum,also belongs to the circle of problems traditional for approximation theoryin ergodic theory.5. The second V.A.’s PhD student in ergodic theory was R. M. Be-linskaya (Ekhilevskaya, 1938–2011). Her main topic was time changes inautomorphisms and related problems of ergodic theory. The results of herthesis (1970) are widely used in the literature. The main problem suggestedto her by V.A. will be described below. But first it is worth mentioning thatV.A., at our request, made several surveys of open problems in ergodic the-ory, first time around 1962, and then in 1965 (mostly about entropy-relatedproblems).V.A. mentioned various problems, but all of them were related to hisown work. In the first talk, he promoted the study of endomorphisms andsuggested to classify finite monotone sequences of measurable partitions astheir invariants. It is appropriate to mention, in particular, O. V. Guseva,a participant of the seminar during its early years, who worked mainly onthe theory of partial differential equations, but attended the ergodic semi-nar, took up this subject, and, generalizing the classification of measurablepartitions given by V.A., obtained a complete classification in the case offinite sequences.Another problem was especially important: V.A. asked when two ergodicautomorphisms have isomorphic orbit partitions. He said that this questionappeared in his studies in the 1940s, but it was not known even whether this10s true for two nonisomorphic rotations of the circle. V.A. himself believedthat in this case the answer should be positive. And his bold conjecturewas that if two automorphisms have different entropies, then their orbitpartitions are nonisomorphic. It is this problem that R. M. Belinskaya sys-tematically tried to solve for a long time, discovering interesting facts abouttime changes along the way. In particular, she observed that the orbit parti-tion of an automorphism is the intersection of a decreasing dyadic sequenceof partitions. But the main problem remained open. At the end of 1966,R.M. and S.A. informed me of this fact, and soon I proved that all ergodicdyadic sequences are lacunarily isomorphic, and hence their intersectionsare metrically isomorphic. Thus, V.A.’s conjecture turned out to be false.V.A. was very glad that the answer was finally obtained. The results werepublished in
Functional Analysis and Its Applications in 1968. But whilethe paper was in preparation, I discovered, selecting material for our survey,joint with S.A., on dynamical systems and operator algebras for
Itogi Nauki ,that as early as in 1963, the American mathematician H. Dye proved thisfact in terms of the theory of operator algebras etc. Dye mentioned that thisproblem had been implicitly posed by J. von Neumann, in a paper from thefamous series of joint papers with F. Murray, as the isomorphism problemfor hyperfinite factors of type II . Von Neumann believed that the answeris most probably positive, but that it would require a detailed analysis ofergodic automorphisms. It is this problem that Dye solved. It is worthmentioning that V.A. knew about the isomorphism problem for factors, butdid not know about its relation to his question. Dye’s theorem is a classicalfact of ergodic theory and is much cited in the literature. My proof, relyingon the lacunary isomorphism theorem, is much simpler than the originalone. Of course, these results were reported both at the ergodic and Moscowseminars.At first, it seemed to me that a strengthening of the method of provingthe lacunary isomorphism theorem would allow one to prove the isomor-phism of any two ergodic dyadic sequences. In my paper on lacunary iso-morphism published in Functional Analysis , I even made the correspondingremark. However, very soon I became aware of the difficulty of the problem,and it took me a year before I managed, on the one hand, to find a “stan-dardness criterion,” i.e., a necessary and sufficient approximative conditionfor a given dyadic sequence to be isomorphic to a simplest (or standard)sequence, and, on the other hand, to present a continuum of pairwise noni-somorphic ergodic dyadic sequences (they are now called dyadic filtrations).The corresponding invariant was a new metric invariant, the entropy of a fil-tration. Following the natural logic of the subject, I defined the so-called11main invariant” of a filtration, and somewhat later, the corresponding in-variant of an action, the “scale” of an automorphism. Soon after my resultswere reported at the Moscow ergodic seminar, A. M. Stepin proved that theKolmogorov entropy of the action of the group σ ∞ Z is also an invariant ofa dyadic sequence. The entropy of a filtration introduced by me coincidesin this case with the entropy of the action of the group. Curiously enough,V.A.’s intuition has been eventually almost vindicated: the entropy of anaction is an invariant close to an orbit invariant, but still not exactly an orbitinvariant. The main results were included into my Doctor of Science disser-tation “Approximation in measure theory.” As far as I remember, it wasreported at probably the last meeting of the ergodic seminar (around 1971).Later, this circle of problems (orbit theory, the theory of filtrations, etc.)grew into a large research area and found applications in a vast variety offields: the theory of random processes, asymptotic combinatorics, and, ofcourse, ergodic theory. The theory of filtrations must provide the foundationfor a combinatorial (approximation) approach to dynamics. At the moment,it is far from completed.A few words about other seminar participants.I have already mentioned the young mathematicians M. I. Gordin,A. A. Lodkin, A. N. Livshits, and others, who took an interest in ergodictheory. Among my graduate students of that time working in this field,A. N. Livshits (1950–2008) stood out. He was a highly gifted and earlymatured mathematician. He had caught only the very end of the ergodicseminar, and participated mainly in my later seminar, to which he cameas a first-year student. Very quickly A. N. Livshits mastered the theoryof Anosov systems and, while a third-year undegraduate, proved his fa-mous theorem on the cohomology of hyperbolic automorphisms and flows.Then he worked much on encoding and isomorphism, independently provedKrieger’s theorem (see above), and later we jointly established a direct re-lation between adic transformations and substitutions.By the time of the new flourishing of ergodic theory connected withthe name of D. Ornstein, who solved the isomorphism problem for Bernoulliautomorphisms and discovered non-Kolmogorov Bernoulli systems, the sem-inar had already slowed its activity. These papers were not discussed at theseminar. But it is appropriate to mention that one of the seminar partici-pants R. A. Zaidman, prior to Ornstein’s works, claimed to have proved thatthe entropy is a complete invariant of Bernoulli automorphisms. This claimwas widely known. However, repeated attempts of seminar participants tounderstand R. A. Zaidman’s arguments failed. It is worth mentioning thathe is a very talented and interesting mathematician, and his ideas on en-12oding were later picked up by A. N. Livshits, who greatly developed thetheory of codes.The works of other my graduate students in ergodic theory were donelater.In 1969, V.A. offered me to conduct, together with him, a small er-godic seminar for beginners (first and second year students), and I gladlyaccepted. V.A. only outlined the program of the seminar, but almost didnot attend its meetings. The seminar was educational and was active forone term. Its participants and speakers included the students A. A. Suslin,V. V. Rokhlin (Jr.), A. G. Reiman, V. M. Kharlamov, and many others,who later became well-known mathematicians. By the way, this generation(graduated in 1972) was one of the strongest at the Faculty of Mathematicsand Mechanics in those years (A. N. Livshits, B. S. Tsirelson, A. A. Suslin,V. M. Kharlamov, and others).
7. Schools, conferences, the conclusion7. Schools, conferences, the conclusion