2.1. Moscow State University

Katok attended Moscow schools 69 (first and eighth grades) and 637 (second to seventh)Footnote ‡, and in the 9th grade, he transferred to Moscow School N59, from where the likes of V. I. Arnold and V. P. Maslov had graduated. As a 14-year-old high-school student, he

He wished to enter the Lomonosov Moscow State University before his 10th and final grade. With the help of Aleksei Markushevich, the vice-minister of education of the RSFSR, he obtained permission to take the external examination for high schoolers after the 9th grade (at not quite 16 years old) and so never needed to graduate from high school. Yulij Ilyashenko recalls a newspaper headline ‘Markushevich was struck by the mathematical talent of the youth’. Katok having been an alumnus of ‘mathematical circles’ for high-school children (in his case, supervised by A. Egorov and N. Vasiliev), his next move was to affect his life well beyond his career. Upon entering the university in 1960, he immediately began teaching such a circle himself. Among his students was Svetlana RosenfeldFootnote §, almost three years his junior, also at Moscow School N59, and the daughter of Boris Abramovich Rosenfeld (1917–2008) and Lucy Lvovna DavidovaFootnote ¶ (1919–2020). Boris Rosenfeld was a geometer and

Anatole and Svetlana fell in love and married on June 5, 1965, soon after Svetlana turned 18; the flame of their love burned strongly for well over 50 years of marriage. Their daughter Elena Katok was born in Moscow and is now the Ashbel Smith Professor at the Naveen Jindal School of Management of the University of Texas at Dallas. Their son BorisFootnote † Katok, also born in Moscow, is a Senior Software Developer and the owner of Coconut Tree Software in Reno, Nevada. Daniela (Danya) Katok was born in Hollywood and is an acclaimed soprano in New York City.

Anatole continued to teach in mathematical circles, and it is worth describing the evolution of these as the context of his service as one of six assistants to Eugene Dynkin.

In the 1960s, dynamical systems emerged as an independent mathematical discipline in its own right due to seminal developments in the 1950s and around 1960, such as the KAM theory (Kolmogorov, Arnold, Moser [Reference Arnold16, Reference Kolmogorov233, Reference Kolmogorov, Abraham and Marsden234, Reference Moser255]), entropy (Kolmogorov, Sinai [Reference Katok184, Reference Kolmogorov235, Reference Kolmogorov236, Reference Sinaĭ284, Reference Sinaĭ285]), and hyperbolicity (Smale, Anosov, Sinai [Reference Anosov4–Reference Anosov6, Reference Anosov11, Reference Anosov and Sinaĭ15, Reference Smale288])Footnote † Young mathematicians, such as Gurevich, Margulis, Oseledets, Ratner, Stepin, and Katok, found this rapidly developing theory exciting and attractive, and began to work in it. Katok started his third year under the supervision of Robert Minlos, but his enthusiasm waned by the end of the year. While Minlos was a distinguished mathematician and an inspiring teacher, the problems he assigned in his seminar required a lot of technical skill and did not captivate Katok. Much later, Katok voiced the opinion that Minlos was at his best ‘when he collaborated with others who were able to show him the way’Footnote ‡. For instance, although Minlos started to work on statistical physics earlier than Sinai, Katok believed that real progress began only when Sinai entered the field and gave direction to Minlos’ superb technical skills (to work on phase transitions in the Ising model [Reference Minlos and Sinaĭ253, Reference Minlos and Sinaĭ254]). Thus—and on the advice of Minlos—Katok started attending Sinai’s special course on ergodic theory (K-systems and related topics) in 1963, at the very time Sinai began to receive wide recognition [Reference Mešalkin and Sinaĭ251, Reference Rohlin and Sinaĭ276, Reference Sinaĭ286, Reference Sinaĭ287]. They started meeting and discussing mathematical problems, and Katok worked with Sinai for the last two and a half years at the university as well as in graduate schoolFootnote §. Leonid Bunimovich recalls:

2.2. Two approximations

Katok received his doctoral degree from Moscow State University in 1968, and his dissertation reflected the much-cited work with Stepin [Reference Katok153, Reference Katok152, Reference Katok and Stepin223, Reference Katok and Stepin224] (and Oseledets, see [Reference Dynkin80, p. 7] and the review of [Reference Katok152]), which Sinai had encouraged and supported. This concerns periodic approximations of probability-preserving transformations. A $\mu $ -preserving invertible transformation T of a probability space $(X,\mu )$ is said to allow approximation by periodic transformations with speed $\scriptsize \begin {cases}f(n)\\o(f(n))\end {cases}\!$ if there are finite partitions $\xi _n=\{C_n^k\mid 1\le k\le q_n\}\to \mathcal {E}$ (the point partition) and $q_n$ -periodic invertible $\mu $ -preserving transformations $S_n$ , such that $\scriptsize \sum _{k=1}^{q_n}\mu (T(C_n^k)\Delta S_n(C_n^k))\begin {cases}<f(q_n)\\=o(f(q_n)).\end {cases}\!$ Depending on f, this has various dynamical consequences such as ergodicity, mixing, multiplicity of the spectrum, and entropy bounds. For instance, an ergodic T can be approximated with speed $o(1/\ln n)$ if and only if its entropy $h(T)$ is 0, and if T is ergodic, then

$$ \begin{align*}h(T)\le\inf \{c\ge0\mid T\text{ allows approximation with speed }c/\ln n\}\le2h(T).\end{align*} $$

This has applications to interval-exchange transformations (which in turn are connected to geodesic flows on flat surfaces), square roots of transformations, and flows on $\mathbb T^2$ [Reference Katok152, Reference Katok and Stepin224].

The work helped solve some long-standing problems that went back to von Neumann and Kolmogorov, and it earned Katok and Stepin the prize of the Moscow Mathematical Society for young mathematiciansFootnote †. Katok presented it in a 15-minute communication at the 1966 International Congress of Mathematicians in Moscow. Yulij Ilyashenko recalls that

Also in 1966, Katok took a reading course on algebraic topology with Dmitri Anosov, which included pondering a specific, then unsolved, problem relating topology and dynamics: rationality of the $\zeta $ -function for Anosov diffeomorphisms [Reference Manning245]. Pontrjagin’s favorite student [Reference Katok187, p. 11], Anosov was a dynamicist at the Steklov Institute (or Mathematical Institute of the USSR Academy of Sciences), who played a central role in the creation of the modern theory of hyperbolic dynamical systems, and whose name has since been attached to important classes of such dynamical systems. Their interaction continued, and Katok recalled:

Much later, Katok told Ilyashenko that the ‘best way to overcome a depression is to sit and write a mathematical paper’. The Anosov–Katok ‘approximation by conjugation’ (‘AbC’) method [Reference Katok and Hasselblatt190, §12.6] or ‘method of fast periodic approximations’ [Reference Anosov and Katok13, Reference Anosov and Katok14] produces, just for starters, area-preserving ergodic diffeomorphisms of the disk. This itself is surprising, not least because elliptic islands around generic elliptic fixed points are an obstruction to ergodicity. Some experts called this work ‘the main event of the year in ergodic theory’, and the method has over the decades yielded ever more astonishing examples of dynamical systems [Reference Banerjee24–Reference Banerjee and Kunde26, Reference Benhenda31, Reference Bronšteĭn52, Reference Fathi and Herman87–Reference Fayad and Saprykina91, Reference Gunesch and Katok115, Reference Gunesch and Kunde116, Reference Katok153, Reference Katok156, Reference Katok157, Reference Kunde238–Reference Kunde241, Reference Melbourne and Windsor250, Reference Sadovskaya279, Reference Saprykina280]Footnote †. It is an ingenious tour de force in elliptic dynamics, and its framework owes to the Katok–Stepin theory of periodic approximations. Here is a brief outline of this singular idea (following [Reference Fayad and Katok88, Reference Katok187]). Consider a manifold with a non-trivial smooth $\mathbb R/\mathbb Z$ -action $\mathcal S=\{S_t\}_{t\in \mathbb R/\mathbb Z}$ and an invariant (averaged) Riemannian metric and volume $\nu $ . The desired exotic examples are obtained as $f=\lim _{n\to \infty }f_n$ of $f_n {:=}H_n\circ S_{\alpha _{n+1}}\circ H_n^{-1}$ with $\alpha _n=({p_n}/{q_n})\in \mathbb Q$ and $H_n=h_1\circ \cdots \circ h_n$ , where the $h_n$ are $\nu $ -preserving diffeomorphisms of M that commute with $S_{\alpha _n}$ , which is achieved by defining $h_n$ as a map between one fundamental domain of $S_{1/q_n}$ and another one (or the same) and extending periodically to M. This is implemented by choosing $h_n$ and then picking $\alpha _{n+1}$ close enough to $\alpha _n$ (in particular with large enough denominator) to make this converge, such as $|\alpha _{n+1}-\alpha _n|\le 1/(2^nq_n\|H_n\|_{C^n})$ . Katok at times called this an ‘uncontrolled induction’. The ingenuity is that this way, one makes the $f_n$ converge while the $H_n$ diverge wildly, though with just enough control for the purpose at hand, which centers on spreading the orbits of the circle action around M. In a related solo paper [Reference Katok153], Katok showed that the Hamiltonian $\sum _{i=1}^m\alpha _i(p_i^2+q_i^2)$ with all $\alpha _i>0$ can be $C^r$ -perturbed (for any r) in such a way that on every energy level, the Hamiltonian flow is ergodic and has discrete spectrum. However,

It has been said that the Moscow ‘Dynamical community was somewhat divided between the followers of Sinai and those of Anosov’ [Reference Katok186, p. 19], and Katok once explained that

Katok was drawn to Anosov’s mathematical personality. They shared (with much of the Moscow mathematics community) an interest in the aesthetic side of mathematics and liked to obtain elegant and internally beautiful results. This brings to mind a description by Katok’s Caltech colleague Richard Feynman of mathematicians (at Princeton) as ‘a happy bunch of boys who were developing things, and they were terrifically excited about it’ [Reference Feynman, Leighton and Hutchings99, p. 86]. Those who knew him could plainly see that Katok was having fun doing mathematics for its intrinsic appeal rather than working on problems dictated by extrinsic scientific necessity (see also page 41). Indeed,

2.3. History and politics

Katok’s time at Moscow University fell into the golden age of Moscow mathematics [Reference Zdravkovska and Duren303], and Katok at times explained that his choice of mathematics as a vocation was influenced by the relative freedom mathematicians enjoyed because their discipline was least affected and controlled by ideological impositions. He thought that otherwise he might have chosen to become a historian or a diplomat. To those who knew him, diplomacy seems a less natural fit, but even though he was never shy about his opinions,

Indeed, he had an inclination to statesmanship when it was for a good cause. The Katoks visited Kiev in 2014, and then again in June 2015 after the Russian attack on Ukraine. A planned conference had been canceled, then postponed, because participation had collapsed in the face of news coverage. He encouraged others to go, spoke at the conference, and made a point of staying longer to show support for Ukraine and Ukrainian mathematics. He understood that sometimes just being there goes a long way. Likewise,

Lest the reader wrongly suspect a central-European bias,

History and politics remained a strong interest of his throughout his life, and stories abound about his prowess in that field. Among these are many about his command of the underlying facts. Adam Abrams recalls:

Yakov Pesin recalls a party during which the challenge arose to name all US Presidents. Katok dismissed this as a trivial exercise and proceeded to name them all, add in all vice-presidents and throw in all Roman emperors in order without hesitation, with apologies for not knowing in every case the exact years of their reign. For Mariusz Lemańczyk [Reference Katok and Lemańczyk202], the first conversation

Vaughn Climenhaga remembers one of the conferences in Będlewo when there was a bus excursion, and

Of course, as in mathematics, recall of the facts is helpful, but understanding how to work with them is a higher skill—which Katok possessed in both mathematics and history. Pesin (as do others) recalls a ca. 1970 special lecture given by the well-known topologist Mikhail Postnikov on a ‘new chronology’ in history. Nowadays, an internet search turns up the basic context quicklyFootnote † [Reference Diacu77]: this is a fringe theory to the effect (in short) that the dark ages never happened, that this is the reason for the paucity of pertinent documents, and that we are therefore by centuries closer to Roman times (say) than per the accepted chronology. This theory was developed by Nikolai Alexandrovich Morozov and had fallen into obscurity when Postnikov (and much later, Anatoly Fomenko) took it upon himself to bring mathematics to bear on this problem and ‘rescue’ the theory. Having been preceded by a lecture series at the History Institute of the Soviet Academy of Sciences, his lecture on this subject to this auditorium packed with mathematicians was compelling. A few in the audience had questions. Then Anatole Katok, a lowly junior mathematician and, as such, in the back of the auditorium, rose to say that he had no questions but wanted to make some comments and that this might take some time. He then dismantled in a perfectly professional manner Postnikov’s position on every single fact from the lecture. The audience, having just been persuaded by Postnikov, was now convinced of the contrary and looked at Postnikov for a reply. After a pause, Postnikov said ‘A week ago I gave the same presentation at a meeting of the Moscow Historical Society to professional historians. None of them presented such serious professional comments, and I have to think about them’. Next, the well-known historian Lev GumilyovFootnote † came to the podium and said that he had been invited to present his view, but that his ‘young colleague’ had already said everything he had to say. Katok had great respect for Gumilyov and was very proud of having been called his colleague.

A recent topical project of his was a paper about missed opportunities during the last four centuries for Russia to become (in a sense) a more Western country, whatever that may be, and Domokos Szász recalled that after 1990, Katok had the idea that Hungary should become a constitutional monarchy with Otto Habsburg as king. ‘In retrospect I am much sorry our country did not consider this alternative’.

2.4. Seminars

Regrettably, mathematics in the Soviet Union became less sheltered in the late 1960sFootnote ‡. The mathematics department at MSU saw increased antisemitism and general oppression against liberal thought, and almost no Jews were accepted as either undergraduate or graduate students, or faculty [Reference Brin and Pesin51, Reference Katok186, Reference Katok187]. So, upon his graduation, Katok was recognized as a prominent mathematician and definitely fit for a position either at the Moscow State University or at the Steklov Institute, but that was impossible by now. He instead assumed his first and only appointment in the Soviet Union, as Scientific Research Worker at the Central Economics–Mathematics Institute of the USSR Academy of Science (CEMI)Footnote §, a known ‘haven’ for some mathematicians because it employed those whom others did not. Eugene Dynkin is a case in point: ‘The powers that be did not like Dynkin’s independent thinking, and he had to leave the MSU in the spring of 1968. After a jobless period, he at last got a position of senior researcher in the department of mathematics of CEMI’ [Reference Katok and Kuznetsov201]. CEMI allowed these mathematicians to combine their work on mathematical problems in economics, if any, with research in pure mathematics:

The liberal and progressive character of the CEMI culture and its leadership proved important for Katok’s career and that of others, and its location almost next door to the Steklov Institute was convenient for the collaboration with Anosov. Katok taught topics courses in dynamical systems and ergodic theory. (For Michael Brin and Yakov Pesin, these were the only coursework in dynamics [Reference Katok186]; of course, the Anosov–Katok seminar, which both attended from undergraduate days on, provided a deep and broad education at the cutting edge [Reference Katok186, p. 7f]. One should note that ‘undergraduate’ education in Moscow went for 5 years, of which the last 2 were what in the US would be considered graduate level, leaving students ready to pass US qualifiers in virtually any mathematical subject area.)

Katok had been attending the Sinai–Alekseev Footnote † seminar at Moscow State University regularly from 1963 and continued to do so through 1970 and then intermittently until 1977. In the fall of 1969, he and Anosov started another seminar at the Steklov Institute (which had to move to CEMI in 1975 [Reference Katok186, p. 8f]) with Katok as the driving force. While Katok was rather fortunate in his appointment, at that time, many mathematicians beginning their research careers took jobs in organizations wholly unrelated to mathematical research or the teaching of mathematics. These seminars were at the heart of Moscow mathematical culture and a vital connection to the research enterprise for those laboring in these day jobs outside of mathematics and who were otherwise utter outsiders with no standing at these institutions.Footnote ‡ The role of the seminars was twofold: in addition to talks about research by participants and guests, many talks were about works by other mathematicians, mostly foreigners, very often based on preprints, which were rare and precious. The seminars served as incubators of a staggering amount of mathematics, and here as well, Katok’s generosity in sharing ideas was important yet rarely reflected in the literature. For instance, it was his idea to consider the ‘stadium’ as a candidate for an ergodic non-dispersing billiard system [Reference Katok183, p. 239]. The seminars typically met weekly in the afternoon or evening, with well-defined start times and much less well-defined end times, though officially they were scheduled for some two hours duration. For two decades, the Sinai–Alekseev and Anosov–Katok seminars were the main engine that put Moscow at the forefront of developing the modern theory of dynamical systems. To illustrate the point, the core participants of the Steklov seminar were A. A. Blokhin, M. I. Brin, L. A. Bunimovich, E. I. Dinaburg, R. I. Grigorchuk, A. A. Gura, M. V. Jakobson, S. B. Katok, Y. I. Kifer, A. V. Kochergin, A. B. Krygin, G. A. Margulis, Y. B. Pesin, B. S. Pitskel, R. V. Plykin, E. A. Satayev, and (occasionally) T. V. Lokot, V. I. Oseledets, L. D. Pustylnikov, and A. M. Stepin. The long list does not imply that everyone attended all the time; ‘sometimes in the room were two professors and two students’ [Rostislav Grigorchuk].

Rostislav Grigorchuk retains a characteristic memory from his first research talk, which Anosov and Katok had invited him to give at their Steklov seminar in 1975, when he was still a 5th-year undergraduate at Moscow State University and had proved his first result (the now well-known cogrowth criterion of amenability of groups [Reference Grigorchuk112, Reference Grigorchuk113]). Grigorchuk

Pre-empting the speaker or producing the speaker’s theorem or proof was not usually the point. Quite the contrary, as described by Giovanni Forni from being on the ‘receiving end’ of such treatment:

For many, these comments were most eagerly anticipated. As Amie Wilkinson put it:

Aside from matters of taste, the interest in a piece of mathematics is also related to the context, and he

Indeed, whenever Katok went on the record, notably in his historical and biographical writing, he was quite scrupulous about being fair in his assessments. He told me about the difficulties he faced when writing about Anosov [Reference Katok187]: in his latter years Anosov himself wrote about the history of dynamics with an emphasis on the developments in his lifetime, and it was not easy for Katok to describe these fairly without marring this account by pointing to conspicuous shortcomings, which would have been at odds with the purpose of Katok’s article as well as a distraction from the appraisal of the valuable work Anosov himself had been involved in. Katok was appalled and struggled with this for some time until he seized upon the device to briefly discuss the distortions and omissions in these writings and to refer to Mathematical Reviews (specifically my review of [Reference Anosov10]) for a more detailed discussion of these problems.

Generally, Katok’s active seminar and conference participation is well remembered for the way in which it enriched the experience and value of a talk for both the audience and the speaker. He would comment on the meaning of a result, the context which makes it interesting, prior work on which it builds, and why it is important and interestingFootnote † —and sometimes at length:

Moreover, his presence in seminars strongly reflected both the best of the Moscow seminar culture as well as the way in which Katok himself discovered new mathematics, especially in collaboration:

However, sometimes the speaker virtually disappeared:

Helmut Hofer, who now co-organizes these Oberwolfach meetings, describes these exchanges by telling the emblematic story of the ‘Katok diagonal argument’. Katok’s usual seat was in the front right (sitting in front was his general custom, and the doors being on the right in the main Oberwolfach meeting room makes this the natural side for late-comers). Michel Herman might sit in the very left-rear corner. It was not uncommon for one of Katok’s amendments to someone else’s talk to be countered by Herman from the opposite corner: ‘I can’t believe that I am hearing this! I have three counterexamples to that statement. No, I have infinitely many counterexamples to what you are saying…’, and so the argument across the diagonal of the room was on. This is well-remembered, of course, not only because it was entertaining, but because these interactions were illuminating for those present and emblematic of the creative spark of those meetings—which was in no small part due to Katok.

Likewise, I remember from the classes he taught those distinctive moments when he turned from the board to face the audience and explain what it all means. He did not prepare the technical points of his classes in advance, and the proofs were presented off the cuff, which meant that details could be missing or incorrect. But proofs serve to illuminate, and their ideas as well as the rich context he provided gave a unique picture of the subject.

In my Caltech days, Katok ran a seminar as well as a working seminar; in the latter one, students learned the communal practice of mathematics, and they would often be organized around a topic. I recall presenting the classification of higher-rank non-positively curved manifolds after Ballmann, Brin, Burns and Spatzier [Reference Ballmann21, Reference Ballmann, Brin and Spatzier23, Reference Ballmann, Brin and Eberlein22, Reference Burns and Spatzier58, Reference Burns and Spatzier59], and possibly learning more from the questions I was asked than the preparation I had undertaken. I also do not ever remember him taking notes in a seminar or any other context. In an interview, he confessed to this as a weakness, but to my mind it always reflected two major characteristics. Thanks to his prodigious memory, he did not need notes in the first place, and furthermore, his encyclopedic knowledge and vision of mathematics made every new thing an organic part of an interconnected landscape he consummately inhabited and in which he could locate each item at will.

2.5. Research in CEMI years

Returning to the post-doctoral decade in Moscow, it is striking how Katok’s work quickly spanned the breadth of dynamical systems—and more so than a literature search easily reveals. Beyond the multiple papers on periodic approximation, he produced a substantial body of work on what he called ‘monotone equivalence’ (often called Kakutani equivalence and based on a far-reaching generalization of the concept of time-change in flows; ergodic theorists such as Feldman, Ornstein, Rudolph and Weiss were also working on this at about the same time). Here is one application of this notion.

His student Evgueni A. (Zhenya) Satayev did ‘excellent thesis work on Kakutani equivalence theory that came earlier than a similar project by the giants of ergodic theory, D. Ornstein, D. Rudolph, and B. Weiss, and was only marginally weaker than theirs’ [Reference Katok187, p. 6], [Reference Sataev282]. A 1975 announcement [Reference Katok159] was followed by what was meant to be a complete presentation but by no means exhausted the subject at that time [Reference Katok161] because the editor-in-chief of Izvestija was I. M. Vinogradov, an inveterate antisemite whose approval was going to be required for a paper of the necessary length, but would not be forthcoming [Reference Katok187, p. 7ff]. The published paper consisted of half of the intended paper plus announcements of the intended contents of the second half—which could not be published as a separate paper because Katoks’ emigration was then drawing nearFootnote †. (Previously, Katok and Satayev had worked on the number of invariant measures for a flow on a surface [Reference Katok158, Reference Sataev281], and later, they applied the theory of monotone equivalence to flows on surfaces [Reference Katok and Sataev214].)

An oft-cited elliptic foray and a novel application of the Anosov–Katok method was the striking construction of ergodic perturbations of degenerate integrable Hamiltonian systems [Reference Katok157]: The famed and astonishing Kolmogorov–Arnold–Moser Theorem tells us that for small-enough perturbations of completely integrable dynamical systems, a set of almost full measure will retain the orderly dynamics of the original integrable system—provided that system is non-degenerate. This work established that without the non-degeneracy assumption, one can produce the very opposite of this conclusion with arbitrarily small perturbations. The iconic example is that if the geodesic flow of the standard round 2-sphere is perturbed by the addition of an ‘equatorial wind’, that is, a drift akin to a rotation, the resulting dynamics is a Finsler geodesic flow with two ergodic components, indeed, just two closed orbits (the equator run in both directions). These two works illustrate that several of the high points in Katok’s work are difficult specific constructions and mindboggling counterexamples. But he was always thinking of the big picture, including how a particular result fits in the general scheme of things. In these two cases, this fits into a big question on which he worked to a considerable extent during his career: ‘Is there such a thing as smooth ergodic theory?’ Abstract ergodic theory is a discipline dominated by examples and constructions, and he meant to keep in mind the origins of this field by looking for smooth realizations of these various examples and phenomena. Viktor Ginzburg points to the depth and prescience:

Katok’s entry into what is now (thanks to Katok [Reference Katok183]; see page 48) called parabolic dynamics proved lastingly influential as well: it involved the creation of the (Katok–Zemlyakov) ‘unfolding’ method for the study of polygonal billiards [Reference Zemlyakov and Katok304, Reference Zemlyakov and Katok305]Footnote ‡. To study billiards (the free motion of a particle with ‘optical’—or ‘specular’—reflections in the boundary of a region) in polygons with a view to the presence of periodic or dense orbits, say, one can, instead of tracking reflections of the particle, ‘unfold’ each reflection by reflecting the polygon itself in the corresponding side, which for rational polygons may yield a flat surface whose study in turn illuminates the behavior of the original billiard system. While it sounds simple, this idea was foundational for important parts of parabolic dynamics; the paper is among Katok’s most cited ones, and it permanently put Zemlyakov’s name on the map, who published only one other research paperFootnote †.

Much later, during the the Program in Ergodic Theory and Dynamical Systems at MSRI in 1983–84, Katok influenced this field in an entirely different way.

The theory of hyperbolic dynamical systems made explosive progress in this period, and a significant part of it carries Anosov’s name [Reference Anosov4–Reference Anosov7, Reference Anosov and Sinaĭ15]. Katok naturally also joined this frontier of research, and his second publication (after [Reference Katok154]) in this area marks a strand of writing that was to permeate Katok’s career henceforth. His lectures from a 1971 summer school in Katsiveli on the Black Sea [Reference Katok, Mitropolskiĭ and Šarkovskiĭ155] appeared with others on hyperbolic dynamics [Reference Alexeyev, Katok, Kushnirenko, Mitropolskiĭ and Šarkovskiĭ1], and together, these became the main early Moscow text on differentiable dynamics. Katok’s lectures are a systematic presentation of hyperbolic dynamics based on both studying some of the Western work and on filling the Anosov blueprint [Reference Anosov8] for studying the topological dynamics of hyperbolic sets using shadowingFootnote ‡ (he did so again in the supplement to [Reference Nitecki258], which he had also translatedFootnote § and annotated, and the approach can also be found in [Reference Fisher and Hasselblatt102, Reference Hasselblatt127, Reference Yoccoz300]).Footnote ¶. It is notable that this was a much-needed and influential exposition but also cutting-edge mathematics, and that this summer school occurred a mere three years after Katok’s doctorate.

Searching the literature by authorship alone misses essential contributions of his, however. As the mentor of Brin and Pesin, Katok supported the development of the theories of both partially and non-uniformly hyperbolic dynamical systems [Reference Brin and Pesin51, Reference Katok186]. While in a (uniformly) hyperbolic dynamical system, each tangent space splits into complementary sub-bundles in which the action expands or contracts, respectively, a partially hyperbolic dynamical system includes a third sub-bundle, in which contraction and expansion are allowed but at lesser rates than in the former subspaces. Non-uniformly hyperbolic dynamical systems are just that: they possess expanding and contracting sub-bundles as well, but there is no uniform control of the contraction and expansion rates. While it is natural to expect that pushing the techniques from uniformly hyperbolic dynamics is the right strategy, the technical challenges are substantial.

Anosov was the adviser of record for Brin and Pesin—Katok could not serve as such because this would have combined a Jewish student with an adviser who was also Jewish and at an economics institute rather than a university. According to the accepted mode of operation in the Moscow community, Anosov did not propose a problem or provide much guidance, but he was an important listener, and his comments were important—as were his (and Katok’s) machinations and horse-trading to help Brin and Pesin to defend their dissertationsFootnote † [Reference Brin and Pesin51, Reference Katok186, Reference Katok187].

Katok had proposed to Brin and Pesin that they work out what happens if, in addition to the expanding and contracting sub-bundles, there is a sub-bundle with less contraction or expansion, and he gave them a pertinent paper by Sacksteder [Reference Sacksteder278], [Reference Katok186, p. 14]. From there, they proceeded to develop the theory of partially hyperbolic dynamical systems [Reference Brin and Pesin49, Reference Brin and Pesin50]. Katok and Anosov listened and commented, but did not get otherwise involved. When asked whether this work would be enough for dissertations, Anosov told them that they needed to say who proved what, as what they had would only be sufficient for non-Jewish candidates; Jews would need an additional paper. Brin proceeded to study frame flows as an application of partial hyperbolicity [Reference Brin41–Reference Brin43, Reference Brin and Gromov45, Reference Brin and Karcher46, Reference Brin and Pesin48], and as Pesin was casting about for a project, he went back to a book on differential equations ‘with non-zero exponents’ [Reference Bylov, Vinograd, Grobman and Nemyckiĭ61], which Anosov had given him along with the suggestion that it might be useful. Together with the motivation to extend ergodicity of geodesic flows of compact surfaces from metrics with negative curvature to those without focal points, this was the origin of his theory of non-uniform hyperbolicity [Reference Barreira and Pesin27, Reference Pesin263–Reference Pesin268], [Reference Katok186, p. 17].

His first approach assumed that while the contraction and expansion rates may vary discontinuously with the orbit, the attached constant would be uniform. He presented this to Anosov as an add-on to the Brin–Pesin theory, and Anosov saw right away that this could not be right, sending Pesin back to the drawing board. On the two-hour subway ride home, Pesin figured it out, wrote up a draft at home and called Anosov with the good news. Anosov did not buy it and said that he was not willing to listen to more nonsense, but as a last chance he gave Pesin five minutes to convince him. Eventually, Pesin presented this work in the seminar, and it became clear that this was both correct and important. Pesin then wrote the Uspehi paper [Reference Pesin265] (his third paper and second major one), and he recalls being lost in how to explain his theory of non-uniform hyperbolicity (that is, of systems with non-zero Lyapunov exponents) in the publication. Katok read drafts and discussed them generously, he helped the exposition and with putting this work in context—his vision and perspective were well ahead of Pesin’s at the time and crucial for this work. Yet, despite this deep involvement in the work, Katok refused to be named as a coauthor of what is now known as the Pesin theory. (Indeed, Katok and Pesin have no joint publication. Brin and Katok coauthored two papers, of which one [Reference Brin, Feldman and Katok44] builds on Brin’s dissertation work [Reference Brin41, Reference Brin42].)

This is why Brin and Pesin (and the Mathematics Genealogy Project) consider Katok their doctoral advisor even though he was not allowed to officially serve as such and Anosov’s involvement was deep and crucial as well [Reference Katok187]. Katok’s promotion of this work continued past its completion:

Thus, within a decade from his doctorate, Katok was profoundly influential and internationally known not only in ergodic theory but in (what is now known as) elliptic, parabolic, and hyperbolic dynamics.

3.1. Maryland

In August, the family was picked up at Dulles airport by Sasha Gruz and Dora Katok, who had moved to Rockville a little earlier. After staying with them for several days, the Katoks moved to a rental apartment in College Park and settled in. Svetlana was finally able to pursue her doctorate in mathematicsFootnote ‡ and Anatole, now 34, assumed his first professorship.

A few remarks on some of the works from that period. Several of these owe to the preceding time at the IHES, such as the book on hyperbolic dynamical systems with singularities [Reference Katok, Strelcyn, Ledrappier and Przytycki225]. Two Annals papers produced Bernoulli diffeomorphisms on any compact connected $C^\infty $ manifold, possibly with boundary, of dimension greater than 1. The first was written by Katok at the IHESFootnote † [Reference Katok163] and produces Bernoulli diffeomorphisms on surfaces, notably on a disk. This is profoundly astonishing: these are area-preserving diffeomorphisms which, up to a measurable change of ‘coordinates’, are isomorphic to a Bernoulli shift, that is, the probabilistic model of a fair coin toss (with possibly more than two sides). At the core lies the Pesin theory: when obtained from hyperbolicity in the right circumstances, ergodicity implies the Bernoulli property. The ideas are so natural that they cannot be omitted here. Katok starts with a linear hyperbolic automorphism of the 2-torus with four fixed points. He ‘slows it down’ at the fixed points so these become neutral instead of hyperbolic; the resulting Katok map is non-uniformly hyperbolic but topologically conjugate to the initial automorphism; it has stable and unstable foliations with smooth leaves and is Bernoulli by the Pesin theory. The slow-down is carried out with enough symmetry that the map factors through the branched double covering of the 2-sphere (which is smooth off the four fixed points). Puncturing at the branch point gives a like map on the disk ( $C^\infty $ tangent to the identity at the boundary), which in turn can be identified with the complement of the skeleton of the desired surface. Such Katok maps on the torus (usually with only one fixed point for simplicity and under the name ‘the Katok map’, or, more generally, almost Anosov maps, that is, maps that are hyperbolic except for finitely many neutral fixed points) have been of growing independent interest as a proving ground for the theory of non-uniformly hyperbolic dynamical systems because they lie at the very edge of uniform hyperbolicity and so are more tractable in many respects than other non-uniformly hyperbolic dynamical systems [Reference Alves and Leplaideur3, Reference Hu and Young135–Reference Hu, Pesin and Talitskaya139, Reference Pesin, Senti and Zhang262, Reference Veconi291, Reference Wang296, Reference Zhou306].

This first result in turn provided two steps of four in the proof that the same can be done on any manifold: Feldman and Brin each developed a method for inserting this example into a higher-dimensional manifold (Brin’s method being showcased as being of independent interest), and a sophisticated result by Rudolph provided the isomorphism to a Bernoulli shift of the resulting diffeomorphism [Reference Brin, Feldman and Katok44]Footnote ‡.

Also partially written at the IHES, Katok’s most cited paper by far, Lyapunov exponents, entropy, and periodic orbits for diffeomorphisms [Reference Katok165], produced several celebrated results by combining the shadowing approach of Anosov and Bowen (to whose memory the paper is dedicated—some two months older than Svetlana Katok, he had died just as the Katoks arrived in the US) with the Pesin theory to great effect [Reference Katok and Hasselblatt190, Supplement]. A core idea sounds deceptively simple: an orbit that is Lyapunov-regular in the sense of the Pesin set has enough hyperbolicity to implement the Anosov–Bowen approach. What makes the applications stand out is that they are so simple to state and effective, such as density of periodic points and lower bounds on their number as well as exhaustion of topological entropy by horseshoes. Corollary (4.2): A $C^{1+\alpha }$ diffeomorphism of a compact manifold with a non-atomic invariant Borel probability measure that has non-zero Lyapunov exponents has positive topological entropy. Bringing in periodic points and an important and entirely new definition of entropy (see [Reference Brin and Katok47] on local entropy, a topological version of the Shannon–McMillan–Breiman Theorem and Katok’s second-most cited paper, as well as [Reference Gurevich117, Reference Katok184, Reference Vershik294]) now produced effective lower bounds on their growth rate and, as Corollary (4.3), the most well-known result of the paper: A $C^{1+\alpha }$ diffeomorphism of a compact surface with positive topological entropy contains a horseshoe [Reference Fayad and Zhang92]. He was more proud of this work than anything else and lectured on it in Brazil [Reference Katok168], the 1982 Rufus Bowen Memorial Lectures at BerkeleyFootnote †, as well as at the 1983 International Congress of Mathematicians in Warsaw [Reference Katok173]. Indeed, the latter address announced significant extensions to appear in a forthcoming publication on ‘Lyapunov exponents, entropy, hyperbolic sets, and $\epsilon $ -orbits’. That publication did not materialize soon or in the intended form, but the subject remained on Katok’s mind. It appeared in a Caltech topics course during the academic year 1986–1987, which was attended by Leonardo Mendoza. The two of them embarked on the project of a proper self-contained account of this within one of non-uniformly hyperbolic dynamics. Due at least in part to Mendoza’s move to Venezuela and a subsequent career change, this project stalled, but not before Mendoza extracted from it in the early nineties a shorter text, which in part concentrated on the two-dimensional context. I remember that Katok showed this to me at the time and was at a bit of a loss as to what to do with this manuscript. It occurred to me that its size was comparable to that of the chapters of ‘Introduction to the Modern Theory of Dynamical Systems’, which was just nearing completion, and we soon decided to include it as a supplement. Its introduction explains what might have been:

While no Katok–Mendoza book along these lines ever materialized, these notes, though themselves now apocryphal, became part of the canon nonetheless. They ‘served as the basis for the first draft of Chapters 1–5 and parts of Chapters 6–7’ [Reference Barreira and Pesin27, Foreword] (that is, close to half) of what is now the standard work on the subject. Katok also ‘fully participated in designing the content of the book in its present form’.

From about the time of the Katoks’ move to Maryland, Keith Burns recalls:

Here, Katok considered geodesic flows [Reference Katok169, Reference Katok175] and gave effective lower bounds on the growth of the number of closed geodesics on a surface, continuing on to what is now called entropy-rigidity for surfaces. If a Riemannian surface has negative Euler characteristic E and volume V, then $\sqrt {-2\pi E/V}$ lies between the topological entropy and the Liouville entropy of the geodesic flow, and when equality holds on either side (in particular, when both entropies coincide), the metric has constant curvature. With Ralf Spatzier, Katok went on to compute both entropies for locally symmetric spaces in arbitrary dimension and found that they agree in all cases. The resulting table is followed by the comment that

Before long, this understated aside became known as the Katok Entropy-Rigidity ConjectureFootnote †. As Spatzier tells it:

The conjecture is still open, and rigidity theory more broadly has been active ever since, thanks in no small part to Katok himself. He not only worked on this conjecture himself but put much effort into popularizing it by giving talks, organizing a series of conferences over two decades, and directing the interest of mathematicians towards this conjecture and related ones. Rigidity theory was becoming a veritable industry, and has been going strong since; among the high points with respect to this particular conjecture are an early paper by Flaminio that made progress but also showed up a material challenge [Reference Flaminio103] (hyperbolic manifolds do not maximize Liouville entropy, even locally), the result of Besson, Courtois, and Gallot that the topological entropy of the geodesic flow is minimized only for locally symmetric metrics [Reference Besson, Courtois and Gallot36, Reference Besson, Courtois and Gallot37], and the extensions of Katok’s result to contact and then volume-preserving Anosov 3-flows by Patrick Foulon [Reference Foulon105] and De Simoi, Leguil, Vinhage and Yang [Reference De Simoi, Leguil, Vinhage and Yang283]. Lyapunov-exponent rigidity [Reference Butler60] is closely related, and early on, this program and the rigidity conferencesFootnote † also developed strong interactions with the Zimmer program [Reference Fisher100, Reference Fisher101, Reference Katok177].

True to form, simultaneously with these papers on hyperbolic dynamics, Katok did notable work on both elliptic and parabolic dynamics as well as ergodic theory.

In elliptic dynamics, he worked on variational methods [Reference Bernstein and Katok35] and gave a new proof of Mather’s then-recent theorem about the existence for twist maps of quasiperiodic orbits of all frequencies in an interval [Reference Mather247], which went back to Birkhoff’s original methods. This produced further insights into these Mather sets [Reference Katok171]. (A yet different method had some two years prior to Mather led Aubry to like results [Reference Aubry and Le Daeron19]—with different motivation and terminology—and the invariant Cantor sets have come to be called Aubry–Mather sets.) Katok and Mather thought about a joint paper that would combine their different methods, but this only resulted in unpublished 1982 preprints (More about Birkhoff periodic orbits and Mather sets for twist maps and Continuation of the preprint ‘More about Birkhoff periodic orbits and Mather sets for twist maps’) by Katok. Some of that common perspective is discussed in a proceedings paper [Reference Katok172]; see also [Reference Mather249] and [Reference Katok and Hasselblatt190, Chs 9 and 12].

A study of interval-exchange transformations and polygonal billiards [Reference Katok164] established the absence of mixing: no interval-exchange transformation is mixing with respect to any invariant Borel measure. The same goes for special flows built on these under a function of bounded variation. Therefore, no flow of a rational polygonal billiard is mixing on any level set of the angle function.

Katok continued his quest to understand which phenomena from ergodic theory are realized in smooth dynamics [Reference Katok166]: there are diffeomorphisms with the Kolmogorov property which do not have the Bernoulli property. Even in a purely measure-theoretic context, it is not a simple exercise to find such an example.

The ability to travel internationally and invite visitors to Maryland from all over the world made a real difference in what Katok was now able to do; this can be seen in the acknowledgments in [Reference Katok169, pp. 343–344] and [Reference Katok, Strelcyn, Ledrappier and Przytycki225, p. VIII], as well as the very existence of [Reference Katok and Kifer196]Footnote †. Indeed, Katok loved traveling and collaborationFootnote ‡ and over time, held visiting appointments (of a month or more) at Cambridge University, ETH Zürich, Erwin Schrödinger International Institute for Mathematics and Physics (Vienna), Federal University (Mexico City), Hebrew University (Jerusalem), IHES (Bures-sur-Yvette), Instituto de Matemática Pura e Aplicada (Rio de Janeiro), Independent University of Moscow, Institut Mittag-Leffler (Stockholm), Japan Society for the Promotion of Science (Invitation Fellowship), Mathematical Sciences Research Institute (Berkeley), Mathematical Institute of the Polish Academy of Sciences (Warsaw), Mathematics Research Center of the University of Warwick, Chinese Academy of Sciences Morningside Center of Mathematics (Beijing), National University of Uruguay, Nicolaus Copernicus University (Toruń), SFB ‘Geometrie und Analysis’ (Göttingen), Stanford University, Stefan Banach Center (Polish Academy of Sciences, Warsaw), Tata Institute for Fundamental Research (Bombay), Tsing Hua University (Taiwan), Université de Grenoble I, University of Paris VI, VII, XIII, University of Rome, University of Rome II, Weizmann Institute of Science (Rehovot, Israel), and Yeshiva University.

In the US, Katok was also in a position to build a research group. At Maryland, he found Joseph Auslander, Kenneth Berg, Nelson Markley, Walter Neumann, Daniel Sweet, Scott Wolpert, and James Yorke, and during his time, the department was joined (among many others) by Michael Brin, Daniel Rudolph, and Marlies Gerber, a former student of Feldman’s, who came as an NSF postdoc [Reference Gerber and Katok109]:

Katok also immediately took on doctoral students, E. A. Robinson [Reference Arthur Robinson274] and Charles Toll Footnote ¶ being the first (plus, for a time, Chris Hall), as well as Ralf Spatzier (whose advisor of record was Caroline Series at the University of Warwick). Collectively, Changguang Dong, Alena Erchenko, David Hughes, and Daren Wei, who were Katok’s students in 2018, expressed some of their memories:

As case studies of supporting his students well past their doctorate, one can consider the biographies of Michael Brin and Yakov Pesin, and the ways in which Katok played an instrumental role in their lives [Reference Brin and Pesin51, Reference Katok186] while and after they got their doctorates (see page 19). When Brin and Pesin graduated from the university in 1970, they were not even recommended to the graduate program called ‘aspirantura’ because of the new political headwinds. Brin got a junior position at the economics research institute of Gosplan, considered a good job. He could have made a career there, including a PhD in economics. Unlike at CEMI, this was a real economics job, though, and Brin did not greatly enjoy it [Reference Katok186, p. 13]Footnote †. So he asked Katok a couple of years in, whether if he went into mathematics, there would a place and a job for him. Katok said yes—something one might consider almost foolhardy in the circumstances:

While this ‘saved’ Brin for mathematics, it did not provide a mathematics job in Moscow after the dissertation defense. In 1978, Katok arranged an official invitation for the Brins from fictitious relatives in Israel, which they received that summer, and which was required for permission to emigrate. They used this invitation to apply for permission to emigrate to Israel in September 1978, and Brin was fired within a month. He recalls eking out a living with odd jobs, such as translating works by Boris Rosenfeld into English, until they left for Vienna in June 1979, from where he took up an invitation to the IHES arranged by Katok. This helped the visibility of Brin’s work and resulted in a collaboration with Gromov Footnote † [Reference Brin and Gromov45]. Katok furthermore utilized the upcoming special year in Maryland to get Brin hired there on a visiting (later permanent) positionFootnote ‡. Indeed, when Katok

Brin had never before held an academic appointment or taught, and he remembers Katok’s mentoring in these new ventures.

Like many others, the Pesins were less fortunate and did not receive permission to emigrate before the doors slammed shut for a decade as the war in Afghanistan started in 1980—and Pesin was stuck in a job even less desirable than Brin’s (at the Industrial Institute of Optical and Physical Measurements; meanwhile, Natasha, Brin’s wife, was a senior editor in the division of mathematics at the ‘Education’ Publishing House). Joel Lebowitz invited Pesin to visit in 1988 (knowing full well that the Soviet authorities would typically deny travel permits for conferences ‘because there is not enough lead time’, he had made it an open invitation, and thanks to this and glasnost, Pesin was actually allowed to go). In what sounds like a trope from crime fiction, the IHES allowed guests one free call outside Paris, and Pesin used this to call Brin in Maryland—they had not talked for a decadeFootnote §. Pesin spoke of wanting to emigrate, and Brin said he should send a CV. Pesin sent whatever he thought that meant, and between them, Brin and Katok made it usable and sent it to Bob Zimmer, then chair of mathematics at the University of Chicago. When the Pesins had received permission to emigrate and started preparations, the University of Chicago called to say that Pesin had a (temporary) job there starting January 1, 1990, and asked when they were going to arrive. They had no idea but in the end made it there by January 23, finding a month’s salary waiting for them, and support from the Zimmers well beyond the call of duty. Along the way, Pesin had been invited to talk in Karl Sigmund’s seminar in Vienna (whom Brin or Katok had contacted), where he also met Josef Hofbauer and Mitchell Feigenbaum; Sigmund’s financial support made them ‘wealthy’ enough to even support some indigent emigrants in turn. Then Laura Tedeschini-Lalli called from Rome (at the instigation of Brin) with an offer of a two-month appointment—and it turned out that they spent three.

From Chicago, the Pesins visited the Katoks (and Caltech) in February—at the very time Katok was negotiating with Pennsylvania State University, and by early March there was an offer for Pesin of a permanent position at Penn State. Since several other institutions were thinking about making him an offer, he consulted with Brin, who told him to go to Penn State: ‘Tolya will be there, and he’ll organize everything, and you can just keep doing mathematics’. So it was, and now Pesin directs the Anatole Katok Center for Dynamical Systems and Geometry.

Academic appointments in the US offered entirely new outlets for Katok’s legendary organizational and collaborative energies. While he always maintained intense seminar activities, upon his arrival in the US, he immediately set about organizing special years such as 1979–1980 at Maryland [Reference Katok167, Reference Katok170], which contributed to cementing early on his reputation in the West as having boundless energy, and 1983–1984 at the Mathematical Sciences Research Institute (MSRI)Footnote †. Later he organized special semesters at the Banach Mathematical Center in Warsaw 1995 and at the Newton Institute in Cambridge 2000, as well as a well-remembered 3-week AMS Summer Institute on Smooth Ergodic Theory in Seattle 1999. (Several of these events produced substantial proceedings volumes.) He was able to travel extensively, and rare was the year in which he did not visit multiple continents.

3.2. Ergodic Theory and Dynamical Systems

Among the early trips were visits to Warwick in 1979 and 1980, and he and William Parry there noted that while dynamics as a research discipline had very much come into its own in the past decade, research publications by dynamicists in the West were scattered across a variety of journals unlike in the Soviet Union, where the important papers could be found among a few journals of record. They saw the need for a unifying voice in this vigorous emerging field, and

The way Katok recalls it,

Katok published far more papers in Ergodic Theory and Dynamical Systems than in any other journal, and he maintained his association throughout: Executive Editor 1981–1987, Editorial Board 1988–1993, Survey Editor 1994–2011, Editor 1997–2011, Editorial Board from 2011Footnote ‡.

A quarter-century on, he did it again and started the Journal of Modern Dynamics with a view to being a highly selective dynamics journal with a broad focus and a democratic editorial boardFootnote †. Giovanni Forni succeeded him as editor-in-chief:

3.3. California

Svetlana obtained her doctorate from the University of Maryland in 1983 (with Don Zagier [Reference Katok228–Reference Katok230]), whereupon the Katoks departed for a year in Berkeley, where Svetlana was a lecturer at UC, while Anatole was at MSRIFootnote ‡ as one of the organizers of a special year at MSRI; in 1990, he described its import as follows.

Indeed, the first of the dozens of aforementioned rigidity conferences occurred as part of this special year. It

The need of the Katoks for two jobs in one region was no secret, and Barry Simon initiated an offer of a professorship for Anatole from Caltech, where they moved in 1984, with Svetlana having a 2-year adjunct position at UCLA (presumably engineered by W. A. J. Luxemburg, then chair at Caltech). For their first time, they did not live in a capital city.

Now 40, Anatole settled into this position, while over time, Svetlana held positions at more than half of the University of California campusesFootnote †. From this move onward, Katok maintained a continuous stream, if not torrent, of doctoral students, 44 in the course of 45 years (four in the Soviet Union, three in Maryland, seven at Caltech, the rest at Pennsylvania State University)Footnote ‡. For most of us, this afforded a communal experience not entirely unlike what the Moscow seminars must have been like, and this was not only due to the multiplicity of doctoral students at any one time but also Katok’s ability to further build a research group by hiring postdocs and faculty, inviting visitors, and drawing on interactions with related disciplines.

A few recollections from my early career may illustrate this. Having come to the University of Maryland as a physics student on a Fulbright scholarship, I happened to live literally around the corner from Laura Tedeschini-Lalli, then a student of Jim Yorke’s, who told me that ‘someone called Katok’ was going to give a course on classical mechanics in the Spring of 1983. That semester, I decided to become a mathematics doctoral student at Maryland, and in the year 1983–1984, took first-year courses plus dynamical systems from Michael Brin and ergodic theory from Dan Rudolph. In April of 1984, I got a letter from the other coast (email would only become widely available later). Katok wrote from MSRI (then at 2223 Fulton Street) that he was moving to Caltech and I was invited to join him there. He cautioned that this much smaller department might feel different from Maryland, because

When I started there, Mark Muldoon [Reference Muldoon257], who had intended to be a student of Barry Simon’s, also began to work with Katok, and soon after, so did Renato Feres [Reference Feres93], who arrived in 1985 from São Paolo. Before long, the number of doctoral students in dynamics rose, and Caltech saw a steady stream of high-caliber seminar speakers and medium-term visitors from around the globe, such as Shmuel Friedland (Visiting Associate 1984), Jean Bourgain (Visiting Associate 1985–1986), Carlos Gutierrez (Visiting Associate 1985–1986), Gerhard Knieper (Bateman Instructor 1985–1987), Howard Weiss (Chaim Weizmann Research Fellow 1986–1989, NSFpostdoc 1989–1990), Hillel Furstenberg (Sherman Fairchild Distinguished Scholar 1986–1987), Jane Hawkins (Visiting Assistant Professor 1986–1987), Leonardo Mendoza-D’Paola (Visiting Associate 1986–1987), Livio Flaminio (Bateman Instructor 1987–1989), David Ruelle (Sherman Fairchild Distinguished Scholar 1987–1988), Endre Szemerédi (Sherman Fairchild Distinguished Scholar 1987–1988), Ursula Hamenstädt (Assistant Professor 1988–1990), Masahiko Kanai (Bateman Instructor 1988–1989), James Lewis (Bateman Instructor 1989–1990), Peter Sarnak (Sherman Fairchild Distinguished Scholar 1990), Mark Pollicott (Associate Professor 1990), among othersFootnote †, plus an active working seminar and almost annual courses on dynamics by Katok, never twice the same. This included some conferences organized at Caltech, of one of which I have a rather unusual memory. On one morning of the third rigidity conference (titled Dynamical Systems Workshop) in March 1988, I sauntered into the department just in time to hear the first talk, when Katok approached me and told me that the speaker had not shown up. ‘Can you talk about your result?’ he asked. I have since consistently had more than five minutes notice when invited to speak. But I had already given seminar talks about the same subject, and this was a first opportunity for me to present it publicly, as well as a good exercise towards developing a central life skill for an academic: to appear to be prepared. I was also given support for conference travel every year (Oberwolfach 1985, Warwick 1986, Maryland 1987, Durham (UK) 1988, Phoenix 1989, Boulder 1989, Oberwolfach 1989), and for several summers in Göttingen.

From the Caltech days, a few larger mathematics projects are worth singling out. With Knieper, Pollicott, and H. Weiss, Katok carefully studied the dependence of entropy on perturbations of Anosov flows [Reference Katok, Knieper, Pollicott and Weiss197–Reference Katok, Knieper and Weiss199], and with Hurder he deeply connected dynamics with foliation theory [Reference Hurder and Katok140–Reference Hurder and Katok142]. Their study of the Godbillon–Vey class became seminal in an initially unintended direction: it necessitated a more careful study of the (transverse) regularity of the invariant foliations of an Anosov 3-flow, which led to another rigidity conjecture: that an Anosov flow is smoothly conjugate to an algebraic one if these invariant foliations are transversely $C^2$ . In Japan, Kanai then almost immediately made a key construction (the Bott–Kanai connection) that enabled him to prove the first such result for geodesic flows on manifolds of dimension greater than two [Reference Kanai150]. He then came to Caltech as a Bateman Instructor.

Katok and Feres began to build on Kanai’s insights and embarked on the chase of the comprehensive such result [Reference Feres94–Reference Feres and Katok96] (see also [Reference Flaminio and Katok104]). And a chase it was: they were aware that in Paris, Patrick Foulon and François Labourie, soon joined by Yves Benoist, were onto the same target. The Feres–Katok strategy was to buttress Kanai’s line of argument with ever more sophisticated dynamical refinements to progressively weaken the needed curvature-pinching hypothesis. Some of this work took place during a summer in Göttingen (with Manfred Denker as host), and I remember the exhilaration of progress as well as the competitive spiritFootnote †. In the end, Benoist, Foulon and Labourie got there first. While it was a consolation that the French team had gotten there using a ‘big hammer’, the open-dense orbit theorem of Gromov, they had also far exceeded the target by treating contact Anosov flows, that is, they obtained rigidity without assuming even the topological structure of a geodesic flow in the first place [Reference Benoist, Foulon and Labourie32–Reference Benoist and Labourie34]Footnote ‡.

The Anosov cocycle and smooth rigidity theory, as well as the later higher-rank rigidity program also motivated normal-forms theory as a through-line [Reference Delatte76, Reference Guysinsky120].

Only after the Caltech days did a project end that had languished as a 1988 preprint [Reference Katok179], in which Katok produced general criteria for ergodicity and the Bernoulli property for volume-preserving diffeormorphisms and flows by ‘prepending’ the Pesin theory with a suitably general version of Wojtkowski’s method of eventually strictly invariant cone families [Reference Wojtkowski299] and then appending the conclusions of the Pesin theory with a uniformity criterion for the cone families that ensures enough stretching of invariant manifolds to go beyond open ergodic components and achieve ergodicity and indeed the Bernoulli property. (These criteria imply, for instance, that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli and that every compact 3-dimensional manifold has a smooth Riemannian metric whose geodesic flow is Bernoulli.) Keith Burns visited Katok at the IHES for a week in 1991 to get this preprint into publishable form:

Of course, the two had already collaborated in a somewhat similar way during the 1984 MSRI semester (with substantial contributions from Ballmann, Brin, Eberlein, and Osserman).

Billiards, by the way, continued to be Katok’s ‘playground’ [Reference Katok183] in a broad way. He studied polygonal billiards further, both alone [Reference Katok174] and with Eugene Gutkin, who was then at the University of Southern California [Reference Gutkin and Katok118], a collaboration that was to be followed up years later with remarkable joint work on convex billiards [Reference Gutkin and Katok119] that implemented some ‘reverse KAM theory’ as follows. KAM theory implies that a strictly convex billiard has (many) caustics near the boundary [Reference Lazutkin242], but Mather showed that there are no such invariant circles if the curvature of the boundary vanishes at any point [Reference Mather248]. This paper quantifies that result to show ‘that if the minimal curvature of the billiard table (appropriately normalized) is sufficiently small, then the convex caustics are located only near the boundary of the table. In particular, we estimate from below the area… which is free of convex caustics… . To interpret our results, let us consider the following ‘mental experiment’.… Suppose that… the minimal curvature of [the table] is gradually decreasing to zero, while the global shape of the table remains essentially unchanged. Then… the convex caustics are gradually pushed out to the boundary’. The paper contains corresponding results for outer billiards as wellFootnote †, and in this case they establish that if there is a point with small radius of curvature, then an annulus surrounds the table that is free of caustics (there will be plenty near the table and others near infinity); the largest such annulus is a Birkhoff region of instability bounded by ‘the last outer caustics of their kind’. This paper further produces interesting examples of caustics, notably, non-convex ones, a significantly more challenging subject.

4.1. Mentoring and collaboration

Throughout, Katok vigorously recruited students, and the vast majority of his doctoral students graduated from Pennsylvania State University. Andrey Gogolev [Reference Gogolyev111] is still amused by how their formal relationship began:

David Hughes recalls how such a first conversation might then go.

This clearly lays out one way in which Katok’s approach diverged from the prevailing practice in Moscow that left students largely to their own devices when seeking a dissertation problem and an approach to it; he was always ready to suggest problems in which he thought a student might be interested, multiple ones if need be, and had a sense of what the outcome would be and what approaches might work. I believe he told me more than once that when a student was working a problem he had suggested, faith (and perseverance) were all that was going to be required. He was invariably generous with his time—I remember only one single occurrence when I came to his office unannounced and he did not immediately drop everything and spend as much time as needed. (And that one time he simply asked whether I could return an hour later instead.) On the other hand, he was also quite willing to give students their own space as described by Spatzier, who recalls that Katok

Katok also delighted in being surprised by new ideas. I recall him taking me from Caltech to a USC seminar in which Jean-Pierre Otal presented his proof of marked length-spectrum rigidity [Reference Otal260, Reference Otal261, Reference Wilkinson298]. I did not know that this was a breakthrough on an infamous problem and only registered that this involved an almost simpleminded central idea. But then I saw Katok listening—he was leaning forward, his eyes popping out and his mouth open. On the drive back, he expressed his amazement that none of the big guns working on this had seen this idea. ‘I am glad I wasn’t working on this, because I would be pretty upset now to have missed this idea’Footnote †.

Among Alena Erchenko’s early memories is that Katok told her at the beginning of the PhD program: ‘You will never know as much as I do. Therefore, the best route is to learn something that I do not know’. She and I think he liked when his students were able to prove something that he had thought should be different. In her work with him,

Likewise, my dissertation result involved a geometric ‘threading’ mechanism [Reference Fisher and Hasselblatt102, §9.6] rather than one of a cohomological nature as in any previous work.

Another conversation with a memorable piece of advice occurred in the aforementioned home extension in State College, where he entertained Jana Rodriguez Hertz and Raúl Ures:

His care for his students went well-beyond attentiveness to their mathematical needs and development. Howard Weiss called Katok the most gregarious mathematician he has known, and the frequent social events hosted by the Katoks have been an important center of a community, serving a sometimes urgent need. As Renato Feres wrote to Svetlana:

He supported his students individually beyond mathematics when he saw a need. He helped me move from one end of the Caltech campus to an upper-floor apartment on the other end, and when Cheng-Bo Yue was recovering from a violent assault, he made sure that we all took turns looking in on him. This did not only happen in the vigor of his 40s but to the end. Changguang Dong:

Indeed, it was among his last wishes, and Svetlana’s, that instead of sending flowers, well-wishers donate to a fund in support of extreme medical expenses for one of his prominent collaborators and former students.

As was the case when teaching mathematical circles or running the seminar in Moscow, Katok always went well beyond his students, postdocs, and colleagues as he supported young mathematicians. He was sure to include young mathematicians in the conferences he organized, his active participation drew them more deeply into the mathematics, and he supported many careers: as Svetlana Katok organized his files after his death, she realized that he had, over time, written letters of recommendation for virtually all dynamicists—for many of them on multiple occasions. Sometimes recommendations were not enough—Keith Burns is quite aware how his career might have gone rather differently had Katok been less active about engaging him:

It is worth noting that Keith Burns was neither a student nor a postdoc of Katok’s. Indeed, generally Katok

That it had taken until 1990 for the Katoks to have good jobs for both Anatole and Svetlana can only have helped attune them to the need for the needs and opportunities associated with such circumstances. Giovanni Forni well remembers how much Katok was invested in this:

Boris Kalinin and Victoria Sadovskaya accepted a double offer to the department and are both tenured at Pennsylvania State University.

Even his mathematical engagement alone could work wonders. Andrey Kochergin, his first doctoral student, volunteers that after 20 fallow years, he again started producing (much-cited) mathematics, once Katok started traveling to Moscow in 1999, and he has remained productive sinceFootnote †. That first trip back engendered a newfound enthusiasm for MoscowFootnote ‡ so strong that he frequently returned and at some point thought aloud about buying an apartment there. However, by 2003, and some 10 visits later, the enchantment had worn off, and the trips ceased.

Ralf Spatzier shared some perspective on collaboration style:

4.2. Books

Indeed, he was increasingly happy to let his collaborators take the lead in writing, just as Burns described (page 38). Yet, he wrote a lot. My large collaborations with Katok involved writing both separately and jointly. They mainly took place during the Penn State years. In my first year at Caltech, I had taken careful notes in his dynamical systems course, and I had polished them in the months after. They went untouched for years, but before leaving Caltech, after completion of my degree in 1989, I brought this up since we had previously considered writing a book on dynamics. It helped that Katok had taught dynamics courses virtually annually, and because of that earlier conversation had obtained class notes from some of those courses. I had in mind a book of some 100 pages, maybe in the Lecture Notes series, and this conversation did not greatly refine the plans, but we parted having decided to give it a go. It helped that I was departing for a tenure-track appointment rather than a postdoc. It probably helped even more that we had no idea what we were getting ourselves into.

A year later, I got married in Pasadena, and we newlyweds embarked on a 1-year honeymoon in Europe: I had deferred offers of visits at the IHES and the ETH in order to take up my Tufts appointment, and after two weeks of tourism in Ireland, we settled in for 5 months each outside Paris and in Zürich—poor as church mice, but quite happy. While I was pursuing research projects [Reference Hasselblatt123–Reference Hasselblatt126], the book project also got serious attention, and it acquired more bulk in the following year. Katok and I both long remembered the particularly harrowing project of writing up the proof of the Stable-Manifold Theorem. During the 1992 special year on Lie Groups and Ergodic Theory with Applications to Number Theory and Geometry Footnote †, we both spent two months at MSRI in Berkeley, and it was here that an unwieldy jumble of written material took shape as a book with gaps to be filled. This was a major turning point, and combined with the realization that we had something in the works quite distinctive from anything else, we had the motivation and momentum for the downhill stretch to the finish line and were done in 1994. It says a lot that our friendship survived 800 pages of writing (and, more so, rewriting). Appraisals of ‘Introduction to the Modern Theory of Dynamical Systems’ [Reference Katok and Hasselblatt190] can be found elsewhere, so I quote David Hughes, who was a doctoral student of Katok’s at the time of his death:

During a July 10–31, 1996 visit of mine to Montevideo, which overlapped with a visit by the Katoks (who coincidentally arrived by the same flight), we started a follow-on book project ‘for undergraduates’ (no manifolds and no measure theory, so mostly example-driven) [Reference Hasselblatt and Katok132], which was half the size but was to take twice as long to write.

On the subject of Katok’s numerous books, one should mention two publications that aren’t quite books, but of comparable size. At over 130 pages, the current-research survey with Sinai and Stepin [Reference Katok, Sinaĭ and Stepin216] might have served its purpose as a stand-alone book, and indeed, as previously mentioned, effectively became the Moscow textbook on its subject—although the evolution of ergodic theory within the Moscow school from this point onward may not quite have been what the authors must have hoped at the time:

‘Principal Structures’ [Reference Hasselblatt and Katok131] lays out for the ‘adult mathematician’ the structure of the theory of dynamical systems in 200 pages in order to provide a common background for the other chapters in the two Handbook volumes [Reference Hasselblatt and Katok130, Reference Hasselblatt and Katok133] of over 1200 pages each.

It may be worth calling attention to a perspective that came into being as we worked on this chapter. It took shape July 15–August 15, 1998, while we worked at the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna (and made our anniversary pilgrimage to McDonald’s). We realized that ‘Introduction to the Modern Theory of Dynamical Systems’ [Reference Katok and Hasselblatt190] had not been as clear about differentiable dynamics as it might have been, by taking ‘low-dimensional phenomena’ as the catch-all for non-hyperbolic dynamics. In dynamics generally, the prevailing prominent paradigms of stable and random dynamics [Reference Moser256] were in the foreground, where ‘stable’ is elliptic dynamics, KAM theory, and related phenomena, while ‘random’ is hyperbolic dynamics, positive entropy, etc. Missing between these opposites is ‘intermediate’ dynamics with polynomial growth such as polygonal billiards, interval-exchange maps, and several parts of homogeneous dynamics. In this chapter, accordingly, we developed the framework of viewing smooth dynamics in terms of elliptic, parabolic, and hyperbolic dynamics. Maybe this has since been absorbed into the fabric of the discipline to such an extent as to sound commonplace, but it was clearly not in our consciousness in the early 1990s as we shaped the organization of ‘Introduction to the Modern Theory of Dynamical Systems’. Katok’s March 1999 Moscow lecture [Reference Katok183] was among the early public enunciations of this point of view.

The aforementioned book with Jean-Marie Strelcyn (and the collaboration of François Ledrappier [Reference Ledrappier and Strelcyn243] and Feliks Przytycki) on hyperbolic systems with singularities [Reference Katok, Strelcyn, Ledrappier and Przytycki225] is one of two monographs of his; the other, ‘Rigidity in higher rank abelian group actions. Volume I. Introduction and cocycle problem’ [Reference Katok and Niţică206] with Niţică, was meant to be followed by a second volume with applications: it introduces the subject with definitions, examples, and preliminaries from dynamics and analysis, then studies rigidity of cocycles. This is an enormous contribution to pull together much of the core of over two decades of research, and at this time, one can only imagine the companion volume. Based on notes with his student E. A. Robinson, ‘Combinatorial constructions in ergodic theory and dynamics.’ [Reference Katok182] was decades in coming [Reference Katok181, p. 108]:

It lets the reader learn ‘from a master of the subject, presenting some of his tools’ [Klaus Schmidt]. The first part lays out the approximation and construction of ergodic transformations by periodic processes from the beginnings of Katok’s career and the second one lays out cohomological constructions, including invariant distributions of partially hyperbolic and parabolic systems, and ‘nice’ coboundaries with ‘complicated’ cobounding functions.

Katok was also seminally involved in the writing of the book on non-uniformly hyperbolic dynamics, though ultimately not as a coauthor [Reference Barreira and Pesin27, Preface].

Two more books are worth mentioning—in the context of a characteristic imprint on undergraduate mathematics education the Katoks made at Pennsylvania State University. George Andrews recalls:

For over two decades, the program assembled undergraduate mathematics majors from all over the country for an intensive one-semester immersion of specially designed courses, seminars, and research-oriented projects. Its core principles and structure were inspired by what the Katoks recalled as the strengths of mathematics education in Moscow, but informed in equal part by having seen that system crush fragile egos of young students with great potential. Anatole Katok taught courses in the MASS program multiple times, and on two of the latter such occasions, Vaughn Climenhaga, then a graduate student, was the assistant for the course. Katok’s unique exposition and Climenhaga’s writing chops produced two lovely volumes [Reference Climenhaga and Katok63, Reference Katok and Climenhaga188] best introduced by Climenhaga’s recollections of assisting with these classes:

4.3. Organization

At Pennsylvania State University, Katok was able to build lasting structures to support dynamical systems worldwide, and he did not take long to get started. Since October 18, 1990, the Penn State–University of Maryland Semi-annual Workshop in Dynamical Systems and Related Topics has been repeated at Pennsylvania State University every fall, with a companion meeting at the University of Maryland every Spring since 1992, the meetings being jointly sponsored by the two institutions and regularly supported by the National Science Foundation. Each time, numerous participants travel from one of these institutions to the other, and they are joined by participants from across the US and the globe. This has by now long been such an established institution that individual invitations are no longer needed; everyone knows to await the announcement of the next date, often during the previous meeting, and to then register. This alone puts Penn State on the mental map of every dynamicist in the world.

Out of this conference, faculty hiring, a visitor program, an ever larger footprint in the graduate program, and Katok’s energy and funds grew the Center for Dynamical Systems and Geometry at Penn State, which consists of a large group of faculty members, graduate students, postdocs, and associate members from other institutions. Its program includes a weekly Dynamical Systems Seminar, the Center for Dynamics and Geometry Colloquium, a Working Seminar on Dynamics and its Working Tools, a Dynamics Student Seminar, a special lecture series, graduate and undergraduate courses, and the Penn State–University of Maryland Semi-annual Workshop. It is this center which has been endowed and named in Katok’s honor.

One concern of Katok’s had long been that dynamical systems as a discipline is less visible than it should be. Much earlier, this had been among the motivations that led to the founding of Ergodic Theory and Dynamical Systems. Characteristically, he perceived that this particularly impacts young dynamicists at critical career stages. Together with Michael Brin, he chose to address this. In 2008, Brin endowed an international prize for outstanding work in the theory of dynamical systems and related areas, given for specific mathematical achievements that appear as a single or a series of publications in refereed journals, proceedings, or monographsFootnote †. The goal of the prize is to recognize mathematicians who have made substantial impact in the field at an early stage of their careers (no more than fourteen years from their PhD). The first Brin Prize was presented in 2008 at the Maryland session of the semi-annual workshop, which was dedicated to Brin’s 60th birthday. Until 2017, this prize was awarded every other year, and it became annual in 2018 (presented at Maryland in the Spring in even years, and at the Penn State meeting in odd years). Giovanni Forni gives us a glimpse behind the scenes:

Since 2016, a further annual prize, now called the Michael Brin Dynamical Systems Prize for Young MathematiciansFootnote ‡, recognizes outstanding contributions to dynamical systems made by researchers within a few years of their PhD (within four years after the PhD at the beginning of the prize calendar year).

Together with the countless doctoral students and postdocs who cycled through the department in these three decades, plus faculty hired during his time, whether still at Penn State or not, Katok made Penn State a center of dynamics, and dynamics central to Penn State—with a broad perspective.

Jerry Bona was the department chair at Pennsylvania State University during the first five years of the Katoks’ tenure there:

Another former department chair, Yuxi Zheng, noted that:

And George Andrews adds: