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Fractal Competition
[The March Workshop on Holomorphic Dynamics included a competition.
Twenty-eight computer-generated images were entered and displayed
around the Institute. The winning entry was submitted by Lasse
Rempe of the University of Liverpool. The following note describes
the mathematics it represents.]
This figure shows a detail of the parameter space of exponential
maps E?: z ? exp(z) + ?. The colored regions are hyperbolic
components and represent particularly simple dynamical behavior:
for parameters in these components, almost every orbit converges
to a stable periodic cycle. The grey region, on the other hand,
represents the bifurcation locus: the region of parameter space
where dynamical behavior changes significantly under a small
perturbation of the parameter.
By a theorem of Schleicher, each hyperbolic component has a
distinguished boundary point, which is the landing point of
two parameter rays (certain dark curves in the bifurcation locus).
The figure demonstrates this fact for a hyperbolic component
(in the middle of the picture), and three other hyperbolic components
which bifurcate from it. It was created to illustrate recent
results (to appear in Proc. AMS), which exploit this combinatorial
structure of parameter space to obtain information in the dynamical
plane.
The boundaries of the four hyperbolic components are drawn
by repeatedly using a Newton's method in two variables to find
parameters with a neutral periodic cycle. A similar method is
used to draw parameter rays. The background picture is a combination
of a heuristic which decides whether a given pixel intersects
the bifurcation locus, and a color scheme on hyperbolic components.
Lasse Rempe (Liverpool)
Contest entries
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Contest Winner
Lasse Rempe, University of Liverpool
(**winning image)
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1.pdf** |
2.pdf |
3.pdf |
4.pdf |
5.pdf |
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| Xavier Buff, Paul Sabatier |
1.pdf |
2.pdf |
3.pdf |
4.pdf |
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| Arnaud Chéritat, Paul Sabatier |
1.gif |
1.png |
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| Hiroyuki Inou, Kyoto University |
1.pdf |
2.pdf |
3.pdf |
4.pdf |
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| Sandra Hayes, Technical University
Munich |
1.pdf |
2.pdf |
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John Hubbard, Cornell
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1.pdf |
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| Alexandra Kaffi, Fields |
1.pdf |
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| Tomoki Kawahira, Fields |
1.pdf |
2.pdf |
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| Sarah Koch, Cornell University |
1.pdf |
2.pdf |
3.pdf |
4.pdf |
5.pdf |
6.pdf |
7.pdf |
8.pdf |
| Shizuo Nakane, Tokyo Polytechnic University |
1.pdf |
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| Rodrigo A. Perez, Fields |
1.pdf |
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| Roland Roeder, Fields |
1.pdf |
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| Dierk Schleicher, International University Bremen |
1.pdf |
2.pdf |
3.pdf |
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| Mitsuhiro Shishikura, Fields |
1.pdf |
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