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 → z 2 + c, where |c| < 1/4, in [10] and an earlier result in [12] 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.

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We show that continuation of holomorphic functions needs time at least Ω(n 2 ) in the uniform case, which is optimal up to a polylogarithmic factor. In the non-uniform case, i.e. assuming f to be computable in time O(t) on an open subset of its domain, we show, that continuation of f cannot be robustly done in time o(n · t(n)). This again is optimal up to a polylogarithmic factor.

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