I just learned that Anatole Katok died on April 30, 2018. He was one of the greatest ergodic theorists of our time. He found and proved many new theorems in dynamical systems theory, introduced new concepts and questions and wrote many books, like of what many would consider the “bible” of dynamical systems “Modern Theory of Dynamical systems” written with Boris Hasselblatt. This book has now almost 5000 citations Source.
Katok proved many fine theorems. Many of them are difficult and technical like his work on refinements of Pesin theory done with Strelcyn (a nice mathematician too whom I would meet later). He developed the theory of speed of approximation (Katok-Stepin theory) and sweet genericity results. Spectacular is also his result that one build smooth Bernoulli systems on any surface (a result from 1979). He would later work on systems with higher dimensional time and rigidity theorems.
When I started to work in dynamical systems theory, this was initiated by a very concrete problem in physics. The Störmer problem studies the differential equation belonging to the motion of a charged particle in a magnetic dipole field), which was the topic of my Diploma thesis of 1987 under the direction of Jürgen Moser, I started to learn ergodic theory on my own and discussed it with Moser in weekly meetings. Naive as I was, I thought it would be easy to prove that the Stoermer problem has some mixing components on a set of positive measure. This could be done with Katok-Strelcyn theory which is an extension of Pesin’s theory to systems with singularities. The motion of a particle in the dipole field just qualifies for that theory. The return map to the equator is a twist map with a singularity where the twist strength diverges. The system is non-integrable because the return map to the north pole and the return map to the south pole do not have common invariant curves. It is like the mixer in the kitchen with two whips. One whip alone does not mix things up because it just turns the fluid but two different whips do mix the fluid. It is an example of a composition of two twists. I would try for a dozen years more until the year 2000 to get better methods for estimating Lyaponov exponents of simple systems. This crashed and I reached the end of the rope on that problem as my fellowship at the university of Texas has ended. The work of Katok was important for me also when working on differential equations on torus (a paper with Bert Hof which essentially sits on Katok-Stepin theory) but uses at that time newly discovered spectral theory.
During the time of writing the senior thesis, Moser gave courses on celestial mechanics and calculus of variations and Lanford on hyperbolic systems, which got me interested in billiards. When Katok gave a seminar talk in Bern about billiards sometime in 1987, I traveled to Bern to see him and his talk. I asked him about what is known about the Kolmogorov-Sinai entropy of smooth convex billiards and he told me than that positive entropy is an open problem. (By the way, it is exactly in situations like billiards, where the Katok-Stepin extension of Pesin theory are needed as in billiards (for example if the table is not convex), the map is not smooth. This can already happen for lp billiards. During a SURF project we measured once the entropy. See the paper [PDF]). The talk of Katok confirmed that Lyapunov exponents is a great topic with many open questions. Most of these questions are still open like also the problem about convex smooth billiards with positive entropy. Much of my thesis was also about Lyapunov exponents as they appear in discrete Schrödinger operators and spectral theory. I slithered so into mathematical physics and then became a Taussky-Todd Postdoc at Caltech. There I continued to work also in ergodic theory. Barry Simon had launched a larger resarch program about singular continuous spectrum and proven at that time a wonderful theorem, the “Wonderland theorem”. I exploited that to give some elegant new proofs of known results, among them a result of Katok and Stepin which assures that a generic measure preserving transformation on a probability space is weakly mixing. (The paper [PDF]). The theorem of Simon is very flexible, one just has to establish a dense set with point spectrum and a dense set with absolutely continuous spectrum (in some topological space of operators) and get generic singular continuous spectrum. One can so also prove that a Bair generic translational invariant measure of a multi-dimensional shift has purely singular cdontinuous spectrum or (using work done together with Bert Hof and using a suggestion of Maciej Wojotkowski from the university of Arizona) that a Hamiltonian system (finite or infinite dimensional) with a KAM torus can be perturbed to have a weakly mixing invariant torus. (A PDF of that paper). This shows in an other way (different from Smale Horse shoe or Melnikov type perturbation theory or Mather theory in the calculus of variations [John Mather also passed away recently. Mather was a mathematician who also influenced me a lot as it was at the same time when Moser gave his topics course on calculus of variations, that Mather was there and would report new results. It was sort of seeing as a student how research is exciting as Mather would often come to the lectures to talk right after to Moser about one of his new discoveries. It was not that I understood anything but to see grown up mathematicians become excited like little kids makes you want to do that too]) that Liouville integrability is fragile.
P.S. What does ergodic theory have to do with quantum calculus? I believe that it does a great deal. A probability space is a finite object, which can be realized on compact spaces. The orbits of a dynamical system are infinite but labeled by the probability space which glues them all together. Like discretization, an embedding in a probability space can have a regularization and compactification effect. But unlike discretization (like tight binding approximations), the probability space approach allows to make statements which are true almost everywhere allowing then for example to make statements which are true Baire generically. At Caltech, with Bert Hof, we looked for example at almost periodic cellular automata [PDF], where one can evolve infinite aperiodic configurations with finitely many data (this is a type of quantization) or almost periodic fluids [PDF” which was work during a SURF project at Caltech: one can look at general relativity or Vlasov systems or any other system which can be defined in a periodic setting and generalize it to the almost periodic case, where one has more symmetry. An almost periodic configuration defines the “Hull”, the closure of all translates, which is a compact topological group. In general, this group is larger dimensional than the original space in which the configurations take place. So, with the help of topological dynamics and ergodic theory, one can look at higher dimensional physics in which we only see the actual 3 dimensional space. This is much less clumsy than “wrapping up spaces” like in string theory or the “multiverse” confusions. In a probabilistic setup, the hull of an orbit = space is the actual universe but the physics is the same for each orbit. All the measurable properties one measures in an almost periodic universe are independent of the starting point. These idea has led to a huge machinery in solid state physics for example, where it has also been fueled with the discovery of quasi crystals, point configurations for which translations have pure point spectrum (for that topic, the book of Marjorie Senechal from 1995 is still the best reference, a book I learned about from Bert Hof also at the time we worked together. Bert Hof was of course the expert in that matter. He wrote his thesis about quasicrystals.)