A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions

Abstract

Although 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|<1/4, in [R. Rettinger and K. Weihrauch, The Computational Complexity of Some Julia Sets, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, 177-185, San Diego, 2003] and an earlier result in [N. Zhong, Recursively enumerable subsets ofRqin two computing models: Blum-Shub-Smale machine and Turin machine, Theoretical Computer Science 197 (1998) 79-94] on the recursiveness of the Julia sets of hyperbolic polynomials.The algorithm given in this paper computes Julia sets locally in time O(k⋅M(k)) (where M(k) denotes the time needed to multiply two k-bit numbers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide a set of disks of spherical diameter 2−k so that the union of these disks has Hausdorff distance at most 2−k+2. This allows to give reliable pictures of Julia sets to arbitrary precision.