Hyperbolic Julia Sets are Poly-Time Computable

In this paper we prove that hyperbolic Julia sets are locally computable in polynomial time. Namely, for each complex hyperbolic polynomial p(z), there is a Turing machine Mp(z) that can “draw” the set with the precision 2 −n, such that it takes time polynomial in n to decide whether to draw each pixel. In formal terms, it takes time polynomial in n to decide for a point x whether d(x, Jp(z)) < 2 −n (in which case we draw a pixel with center x), or d(x, Jp(z)) > 2 · 2 −n (in which case we don’t draw this pixel). In the case 2−n ≤ d(x, Jp(x)) ≤ 2 · 2 −n either answer will be acceptable. This definition of complexity for sets is equivalent to the definition introduced in Weihrauch’s book [16] and used by Rettinger and Weihrauch in [13]. Although the hyperbolic Julia sets were shown to be recursive, complexity bounds were proven only for a restricted case in [13]. Our paper is a significant generalization of [13], in which polynomial time computability was shown for a special kind of hyperbolic polynomials, namely, polynomials of the form p(z) = z2 + c with |c| < 1/4. We show that the machine drawing the Julia set can be made independent of the hyperbolic polynomial p, and provide some evidence suggesting that one cannot expect a much better computability result for Julia sets. We also introduce an alternative real set computability definition due to Ko, and show an interesting connection between this definition and the main definition.