Hyperbolic Julia Sets are Poly-Time Computable

• Authors: Mark Braverman
• Venue: Electronic Notes in Theoretical Computer Science (ENTCS)
• Year: 2005

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 M"p"("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,J"p"("z"))2@?2^-^n (in which case we don't draw this pixel). In the case 2^-^n@?d(x,J"p"("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 [Weihrauch, K., ''Computable Analysis'', Springer, Berlin, 2000] and used by Rettinger and Weihrauch in [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA]. Although the hyperbolic Julia sets were shown to be recursive, complexity bounds were proven only for a restricted case in [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA]. Our paper is a significant generalization of [Rettinger, R, K Weihrauch, The Computational Complexity of Some Julia Sets, in STOC'03, June 9-11, 2003, San Diego, California, USA], in which polynomial time computability was shown for a special kind of hyperbolic polynomials, namely, polynomials of the form p(z)=z^2+c with |c|