Constructing an expanding metric for dynamical systems in one complex variable

Details Constructing an expanding metric for dynamical systems in one complex variable Constructing an expanding metric for dynamical systems in one complex variable

2004-05-31

arxiv.org/abs/math/0406003v2

Nonlinearity 18 (2005) 81-100

Mathematics

Dynamical Systems

18 pages. 8 figures. Revised version (intro. restructured, some minor clarifications through rest)

Scientific paper

10.1088/0951-7715/18/1/005

We describe a rigorous computer algorithm for attempting to construct an explicit, discretized metric for which a complex polynomial map is expansive on a given neighborhood of its Julia set. We show construction of such a metric proves the map is hyperbolic. We also examine the question of whether the algorithm can be improved, and the related question of how to build a metric as close to euclidean as possible. Finally, we give several examples generated with our implementation of this algorithm.

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