Computer Work

Computer experiments can be very helpful for understanding complex Hénon maps. My recommendation is to look at the computer work of others and think about it. However, don't be tempted into doing your own computer work until you have thought very carefully about your goals and how you will achieve them. And think very carefully about what mathematics is involved, and what your computer experiment might reveal.

My moral: If I spend an hour spent thinking about somebody else's computer pictures, it is more productive than 20 hours trying to produce my own pictures.

We are mathematicians, and what is important about computer work:

Does it involve a precise mathematicial formulation?
Does it illustrate an existing theorem?
Does it lead us to a new formulation of a result we might be able to prove? (or prove at least part of it?)

Here are sources I recommend that you look at:

In the Class Notes, there is a short section about computer pictures of unstable slices. These pictures were made using this Mathematica notebook.

Ricardo Oliva has done very nice computer work in his thesis. He developed an algorithm that lets him draw external rays in such a way that he can make them "land" at the Julia set J. He has used this to show very interesting new bifurcation phenomena.

Sara Koch has given a very nice writeup of the software SaddleDrop. Unfortunately, this program is rather old and was written in a very hardware-specific way, and now nobody has the computers for which it was written. But Koch's paper gives a clear idea of the principle of the program. One of the basic ideas of SaddleDrop is that it shows critical points within an unstable slice. This same general topic was explored somewhat differently by Ushiki.

In the thesis of Chris Lipa, he used SaddleDrop to present interesting bifurcation phenomena. It would be interesting to develop a more precise description of these phenomena and perhaps prove mathematically that something like that actually happens.

I recently discovered a paper called excursions. This is not about computer work per se, but it shows a number of computer pictures, and you might find it amusing. This paper appeared in the Proceedings of a conference in honor of the Centennial of the birth of Kiyoshi Oka.

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