Studium einer verketteten Twistbbildung im Störmerproblem

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Oliver Knill

This diploma thesis (senior thesis) was written from 13. Mai 1987 to 13. September 1987 at the end of my undergraduate studies at ETHZ. It was a typical Moser topic: a problem with applicability and fundamental connections to developments in dynamical systems theory: ergodic theory, KAM theory, perturbation theory as well as modeling techniques matter.

About the programming: The pictures were printed on my color needle printer on a resolution of 1200x1200 pixels. All the programming in Pascal was done in the 4 months time frame too. I also needed to write a primitive printer driver to printout the graphics. My Pascal program for the dynamical system experiments had a nice GUI interface written in GEM. The typesetting system was a pixel based text editing system "Signum", which had been quite popular in Europe.

About the Mathematics: The thesis contains an original and new proof of Wojtkowskis theorem which states that a dynamical system for which the Jacobean cocycle has an eventually strictly invariant cone bundle is nonuniformly hyperbolic. This condition is actually necessary and sufficient. I naively had hoped that I could prove positive Lyapunov exponents by looking at the cocycle dynamics induced on measures on the projective phase space because that cocycle map has two attractors located on the stable and unstable direction fields of the map. By the hyperbolicity condition the contraction is a consequence Frobenius theorem. In the nonuniform hyperbolic situation, this is a bit more subtle and settled by Wojtkowskis theorem which implies also Oseledec's theorem in that case. I had hoped that this method would generalize to situations like in the Stoermer problem, where one observes a highly hyperbolic dynamics in the phase space too, but where the invariant cone field is only known on part of the phase space and where one could work with return arguments. I had hoped that one could estimate how much gets lost in the small region one can not control. Most of the ergodic theoretical lemmas in the thesis as well as the new proof are actually "garbage" from different approaches in that direction. The Störmer problem looks promising because without having to introduce a perturbation parameter, the twist strength goes to infinity near the singularity. The singularity is weak enough for Pesin theory to apply, but the mixing strength blows up near that singularity. I would the next 10 years work on the same problem again in much more basic situations like the Standard map, try to use methods from quantum mechanics (the Lyapunov exponent is the logarithm of the determinant of a Schroedinger operator), or complex analysis (subharmonic and plurisubharmonic estimates and various Jensen type generalizations based on an idea of Herman) or calculus of variations (uniform hyperbolicity can be obtained with the implicit function theorem using an idea of Aubry and the hope was that a strong implicit function theory would allow to continue that beyond the uniformly hyperbolic situation) to crack the problem, but without success. It would end with a crash.

About the subject [Added Mai 18, 2011]: The van Allen belt is not only interesting for mathematics as a nonintegrable 1 body problem problem or the physics of Aurora Borealis but also militarily as the clip to the right shows. These experiments were mentioned in the thesis because they illustrate the remarkable stability of the particle motion despite the fact that the system is nonintegrable. It is KAM theory in action. Similarly as in plasma physics or planetary motion it shows that nearly integrable systems can look integrable on substantial subsets. These experiments were of military importance because they illustrated the power of EMP. From the movie: "The Argus experiment thought to create and explore trapped radiation in the earths van Allen belts. Detonating 300 miles above the earth, the experiment thought to create a radioactive shield to impede the performance of a Soviet missile attack". Details [PDF].
About the literature: Most of the literature I have found by consulting the science citation index (in actual printed thick books with extremely thin pages which the ETHZ had in a publicly accessible place close to the dome of the ETH building) and the mathematical reviews and Zentralblatt (also in real book form in the mathematics library). I was especially intrigued by the citation index which allowed me to move forward in time: look who has cited in a paper, then get that article and look who cited that paper etc. I spent many, many afternoons in the library in that summer, read in the morning and wrote or programmed in the evening. The ETHZ library was very nice and the staff very helpful and friendly. One day, one of the librarians took me down into the underground stacks below the ETH Zentrum. Not many people have seen this huge labyrinth which is underground below the building. The librarians would communicate there by whistling codes to each other.

About the advising: I would typically meet once a week with Moser who would in the summer only be available for one or two days a week and I had to make an appointment with his secretary, Frau Liselotte Karrer. Sometimes, I had to wait in line with guests from the FIM who would want to have an audience. Moser gave me complete freedom, what to do. I think he originally intended to steer towards KAM or horseshoe type problems but I started to drift into ergodic theory and to explore the entropy problem. Having read his book "stable and random motion", I knew that this was an important question evenso he warned me early on that this question is "subtle".

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