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

In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time. 1

We establish a new connection between the two most common traditions in the theory of real computation, the Blum-Shub-Smale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1

Abstract. It has been previously shown by two of the authors 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.

While most polynomial Julia sets are computable, it has been recently shown [12] that there exist non-computable Julia sets. The proof was non-constructive, and indeed there were doubts as to whether specific examples of parameters with non-computable Julia sets could be constructed. It was also unknown whether the non-computability proof can be extended to the filled Julia sets. In this paper we give an answer to both of these questions, which were the main open problems concerning the computability of polynomial Julia sets. We show how to construct a specific polynomial with a non-computable Julia set. In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Julia sets with computable parameters. Moreover, assuming a widely believed conjecture in Complex Dynamics, we give a poly-time algorithm for computing a number c such that the Julia set J z 2 +cz is non-computable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.

Abstract. A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. 1. Preliminaries In this paper, we will assume that the reader is familiar with the concept of computability of a subset of R n and its applications to Julia sets of rational functions. We refer the reader to our paper [BY08a] and the book [BY08b] for an introduction to computability of functions and sets in R n, as it applies to the study of Julia sets. A detailed treatment of computability over the reals is found in [Wei00]. We will denote fc(z) = z 2 + c, and Pθ(z) = z 2 + e 2πiθ z two parameterizations of the quadratic family. The latter is more convenient in studying quadratics with a neutral fixed point. We denote Jc, Jθ and Kc, Kθ the Julia sets and the filled Julia sets respectively. Suppose, a polynomial fc has a periodic Siegel disk ∆ centered at a point ζ. Consider a conformal isomorphism φ: D ↦ → ∆ mapping 0 to ζ. The conformal radius of the Siegel disk ∆ is the quantity r(∆) = |φ ′ (0)|. A polynomial Pθ with θ ∈ R has a neutral fixed point at the origin. When this point is of Siegel type, we denote ∆θ the Siegel disk around it, and set r(θ) = r(∆θ). For all other values of θ ∈ R we set r(θ) = 0. Informally, the Julia set Jc (or Jθ) is computable if, given arbitrarily good approximations of the parameter c (or θ), a Turing Machine can output images of Jc (or Jθ) with an arbitrarily high resolution. The parameter is provided to the machine via an oracle, which the machine can query with an arbitrarily high precision. In [BY06] we showed that, surprisingly, there exist parameters c for which the Julia set Jc is not computable. In [BY08a] we demonstrated that such parameters can themselves be computed with an arbitrary precision by an explicit algorithm. The practical implications of these results are quite striking: there are computable values of c for which Jc cannot be visualized numerically.

review articles doi:10.1145/2093548.2093569 Embodied and disembodied computing at the Turing Centenary.

Abstract. In this note we give answers to questions posed to us by J. Milnor and M. Shub, which shed further light on the structure of non-computable Julia sets. 1.

We survey an area of recent development, relating dynamics to theoretical computer science. We discuss some aspects of the theoretical simulation and computation of the long term behavior of dynamical systems. We will focus on the statistical limiting behavior and invariant measures. We present a general method allowing the algorithmic approximation at any given accuracy of invariant measures. The method can be applied in many interesting cases, as we shall explain. On the other hand, we exhibit some examples where the algorithmic approximation of invariant measures is not possible. We also explain how it is possible to compute the speed of convergence of ergodic averages (when the system is known exactly) and how this entails the computation of arbitrarily good approximations of points of the space having typical statistical behaviour (a sort of constructive version of the pointwise ergodic theorem).

In this paper we settle most of the open questions on algorithmic computability of Julia sets. In particular, we present an algorithm for constructing quadratics whose Julia sets are uncomputable. We also show that a filled Julia set of a polynomial is always computable.