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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 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". |
