A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

AbstractAlthough numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of non-trivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers.In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z↦z2+c, where |c|