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 bomb 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".
No this is not from a James Bond movie or Austin Powers spoof, it was real insanity.
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".
