Non-computable Julia sets

Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naïve work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor's distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show: Main Theorem. There exists a parameter value c ∈ ℂ such that the Julia set of the quadratic polynomial f_c(z) = z² + c is not computable. The structure of the paper is as follows. In the Introduction we discuss the question of computability of real sets and make the relevant definitions. Further in this section we briefly introduce the reader to the main concepts of Complex Dynamics and discuss the properties of Julia sets relevant to us. In the end of the Introduction, we outline the conceptual idea of the proof of the Main Theorem. Section 3 contains the technical lemmas on which the argument is based. In §4 we complete the proof.