Algorithmic Construction of Hurwitz Maps

Abstract

We describe an algorithm that, given a k-tuple of permutations representing the monodromy of a rational map, constructs an arbitrarily precise floating-point complex approximation of that map. We then explain how it has been used to study a problem in dynamical systems raised by Cui.

Keywords:

• Hurwitz maps
• dessins d’enfant
• uniformization
• Thurston equivalence
• Sierpinski carpets

2000 AMS Subject Classification::

• 30C30
• 65E05
• 11G32
• 37F20

Notes

¹The code is maintained by the fourth-named author, and is available at https://github.com/jakobkroeker/HMAC.

²This is sometimes called Thurston equivalence.