April 18, 2014

Spring 2006
Holomorphic Dynamics, Laminations, and Hyperbolic Geometry

March 7-11, 2006
Holomorphic dynamics Workshop
In celebration of John Milnor's 75th birthday

Org. Cttee: M. Lyubich, J. Smillie, M. Yampolsky


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

Contest Winner
Lasse Rempe,
University of Liverpool
(**winning image)
1.pdf** 2.pdf 3.pdf 4.pdf 5.pdf  
Xavier Buff, Paul Sabatier 1.pdf 2.pdf 3.pdf 4.pdf  
Arnaud Chéritat, Paul Sabatier 1.gif 1.png  
Hiroyuki Inou, Kyoto University 1.pdf 2.pdf 3.pdf 4.pdf  
Sandra Hayes, Technical University Munich 1.pdf 2.pdf  
John Hubbard, Cornell
Alexandra Kaffi, Fields 1.pdf  
Tomoki Kawahira, Fields 1.pdf 2.pdf  
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  
Rodrigo A. Perez, Fields 1.pdf  
Roland Roeder, Fields 1.pdf  
Dierk Schleicher, International University Bremen 1.pdf 2.pdf 3.pdf  
Mitsuhiro Shishikura, Fields 1.pdf  
and some Fractals by
John Milnor
9.pdf 10.pdf 11.pdf 12.pdf 13.pdf 14.pdf 15.pdf 16.pdf
17.pdf 18.pdf 19.pdf 20.pdf 21.pdf 22.pdf 23.pdf 24.pdf
26.pdf 27.pdf 28.pdf 29.pdf 30.pdf 31.pdf 32.pdf 33.pdf
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