On computational complexity of Siegel Julia sets
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Résumé
Abstract. It is known that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. 1. Foreword Let us informally say that a compact set in the plane is computable if one can program a computer to draw a picture of this set on the screen, with an arbitrary desired magnification. It was recently shown by the second and third authors, that some Julia sets are not computable [BY]. This in itself is quite surprising to dynamicists – Julia sets are among the “most drawn ” objects in contemporary mathematics, and numerous algorithms exist to produce their pictures. In the cases when one has not been able to produce informative pictures (the dynamically pathological cases, like maps with a Cremer or a highly Liouville Siegel point) the feeling had been that this was due to the immense computational resources required by the known algorithms. The next surprise came with the discovery by the authors of this paper in [BBY] that all Cremer quadratics (or more generally, rational maps without rotation domains) have computable Julia sets. The non-computable examples constructed in [BY] were Siegel quadratic polynomials, and one would expect the Cremer case to be at least as bad if not worse computationally. The natural question to ask is then whether in those cases in which we know the Julia set is computable, but no good pictures exist, the computational complexity of such a set is indeed high. Here at least, our original intuition seems to be correct: it is shown in the present paper that there exist computable Siegel quadratic Julia sets with arbitrarily high computational complexity. An irritating possibility still remains that some Cremer Julia sets are computationally easy (and we just do not go about trying to draw them in the right way). This, however, seems unlikely. We note that the examples constructed in this paper are the first known cases of Julia sets which are not poly-time computable. The second author [Brv] and independently Rettinger [Ret] have previously shown that hyperbolic
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|---|---|---|
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