Letters to the Editor Deligne/Zimin Initiative to preserving the Russian mathematical that our paper’s result was not new, Support Young Mathematicians school. and, worse yet, that he believes that As Russian mathematicians and We are deeply grateful to Pierre period three does not imply chaos!!! members of the American Mathe- Deligne for his noble initiative, which Ruelle seems to be aware of only the matical Society, the undersigned feel has already done a great deal to help first of our three conclusions: it is their pleasant duty to inform young Russian mathematicians to (I) (All Periods Exist.) For each positive integer k, there is a point fellow members via the Notices of a survive without giving up research. of period k. The Sharkovsky (1964) remarkable philanthropic initiative This initiative is a continuation of Theorem stated by Ruelle is indeed supporting young Russian research the generous international solidar- ity to Russian scientists which the a more general theorem than part (I), mathematicians undertaken by Pierre American Mathematical Society im- but our theorem does not stop here. Deligne and continued by the Russian plemented in the 1990s and is con- (II) (Mixing.) There is an uncount- philantropist D. B. Zimin. In 2004 tinuing today. able set S ⊂ J which satisfies: For Deligne wrote in a letter to one of us: We are extremely grateful to Dmi- every p , q ∈ S with p = q , “I just won the Balzan Prize. Half the try Zimin for his chivalrous support | n − n | = prize amount is for me to spend on lim infn →∞ F (p ) F (q ) 0, of Russian fundamental science, in a research project agreed to by the particular mathematics. D. B. Zimin a n d Balzan Foundation. I believe that one is one of the very few Russian busi- lim sup |F n (p ) − F n (q )| > 0. of the most useful ways to spend this n →∞ nessmen contributing money for the money (500,000 Swiss francs) would (III) (Divergence from Periodic Or- support of Russian mathematics. be for the benefit of the struggling bits.) The above set S can be chosen Russian school of mathematics.” so that in addition, (1) is satisfied for —Yu. Ilyashenko each p ∈ S and each periodic point Together with several collabora- Moscow Independent University, q ∈ J . tors of the Independent University Steklov Math. Institute of Moscow, Deligne implemented this Dyson’s elegant statement refers to (III). There is also a large literature idea by organizing the “Pierre Deligne —A. Sossinsky that recognizes (II) and refers to it as Contest for Young Mathematicians”, Moscow Independent University, “Chaos in the sense of Li and Yorke” a yearly individual competition of Institute of Mechanics research projects for young Russian, or “Li-Yorke chaos”. We support Dy- son’s view that the definition of chaos Ukrainian, and Byelorus mathemati- —A. Vershik cians, whose laureates are granted depends on the context; that is, it St. Petersburg Branch of Steklov depends on what you know about a a sizable three-year fellowship. To- Math. Institute gether with Victor Vassiliev, Deligne system or what you want to prove. There is a rich tradition of chaos in heads the jury of the contest, which (Received February 20, 2009) topology, where exponential diver- is run along lines similar to those gence of trajectories is not a useful used by the American NSF. Since Response to David Ruelle concept. 2005 Deligne comes to Moscow each Tien-Yien Li and I were delighted I fear that Ruelle’s version of chaos December to supervise the final de- to read Freeman Dyson’s beautiful would leave mathematicians without liberations. During the past four article in the February Notices that a method of dealing with the simplest years, sixteen fellowships have been refers to our paper “Period three im- situations when he asserts “chaos oc- granted, and the money coming from plies chaos” as “one of the immortal curs if the exponential divergence is the Balzan foundation (having in gems in the literature of mathemat- present for long-term behavior, i.e., mind the future payments to recent ics”. We prove one theorem in the on an attractor.” One of the most winners) has been entirely exhausted, paper, and Dyson reports on what famous examples of chaos is the but Deligne intends to continue the we may assume is his favorite part map α − x2 which has a chaotic at- contest by using his personal funds. of it. Assume there is a continuous tractor for a set of α having positive In 2006 the Russian philanthropic function F from an interval J to itself, measure (M. Jacobson). But there foundation “Dynasty” has organized and—to be brief—we assume there is also an open dense set in which the “D. B. Zimin Dynasty Foundation is a “period three” point a ∈ J ; that there is an attracting periodic orbit. Contest for Young Mathematicians” is, F (F (F (a ))) = a = F (a ). Dyson Since it can be shown that there with the same jury and according to simply wrote “An orbit is defined to is only one attractor for this map, the same rules. According to the cor- be chaotic, in this context [emphasis such an α has no chaotic attractor. If responding agreement, this contest added], if it diverges from all periodic Ruelle (who is a superb mathemati- will run for two more years, after orbits.” cian) can prove there is a chaotic at- which it may be continued. The con- But then the June Notices came tractor for α = 3 /2, then I will declare tinuation of these two contests will with a Letter to the Editor by David his definition usable! He simply needs undoubtedly play a crucial role in Ruelle that made two assertions, to prove an infinite set of lemmas: 1232 NOTICES OF THE AMS VOLUME 56, NUMBER 10 Letters to the Editor for each positive integer N, there is Yorke prove that (for suitable maps an alternative perspective of nonneg- no periodic attractor of period N. I of the interval) the existence of a pe- ligible value for discussing the educa- make the following conjecture in the riodic point of period three implies tional opportunities that have arisen spirit of Gödel. a complicated situation reminiscent from the availability of computers. Conjecture. The map 3/2 − x2 has of the homoclinic tangle discovered As will immediately become evident, a chaotic attractor and there is no by Poincaré in his study of the three- what follows is primarily oriented proof of that fact using the usual body problem. Clearly, period three toward instruction through so-called axioms of set theory. implies Li-Yorke chaos. But when “honours” undergraduate calculus I believe that this impossibility of chaos in the solar system is discussed classes, assuming (perhaps some- proof would apply to the large class (Wisdom, Laskar, …), or other ap- what unrealistically) that, in such an of smooth dynamical systems that plications to the real world, another environment, an uncompromising alternate densely between chaotic concept of chaos is used, which refers approach to the subject is permitted. attractors and periodic attractors to “chaotic” behavior for initial con- Most of the relevant memories as a parameter is varied. Ruelle says ditions in a set of positive volume in have their origin in the mathemat- “chaos” should apply only to attrac- the (phase) space where the dynamics ics department of the University of tors. (There are proofs for certain takes place. This condition (rather Capetown and date back to 1934. The very special cases like 2 − x2 .) than uncountability) must be im- offerings in that department included Attractor basins can be fractal— posed because sets of measure zero courses in “applied mathematics”. due to the presence of chaos on an are physically invisible in the present There, you learned to formulate sim- uncountable compact invariant set situation. There has been a semantic ple settings from classical mechanics on the basin boundary. He would shift and, if the map x a x(1 − x) in terms of differential equations, a leave us with no term for such sets!! of [0, 1] to itself has, for some value pursuit leading to inspiring enlight- After Ruelle said that chaos means of a , an attracting orbit of period 3, enment about the true significance “neighboring trajectories diverge ex- most people would not call this map of calculus. In stark contrast to this, ponentially,” he asserted that period chaotic (this is because the orbit of the introductory courses in formal three does not imply chaos! Here is Lebesgue almost every point tends calculus were as discouraging then a compromise. We can define chaos to the nonchaotic attracting period 3 as they are now. in this context as “exponential di- orbit). This being said, the map under In this last context, present educa- vergence for all trajectories on some discussion not only exhibits Li-Yorke tional practice is still operating in a uncountable compact invariant set”. chaos, but also what Yorke at some conceptual desert, being focused on In the spirit of mathematical fun, point described as transient chaos unenlightening manipulative tricks I propose the following conjecture (this is only transient, but visible on of formal differentiation and integra- using this concept. a set of positive Lebesgue measure). tion, the life of many of these being Conjecture. Assume F is a con- As to the very interesting problem supported by the suppression of tinuously differentiable map from an of logical decidability raised by Yorke complex numbers,1 and the need interval J to itself. Then period three (to decide if a point belongs to a set of for most of them having been made implies chaos.