We give a detailed treatment of the “bit-model ” of computability and complexity of real functions and subsets of R n, and argue that this is a good way to formalize many problems of scientific computation. In Section 1 we also discuss the alternative Blum-Shub-Smale model. In the final section we discuss the issue of whether physical systems could defeat the Church-Turing Thesis. 1

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

Although numerous computer programs have been written to compute sets of points which claim to approximate Julia sets, no reliable high precision pictures of nontrivial Julia sets are currently known. Usually, no error estimates are added and even those algorithms which work reliable in theory, become unreliable in practice due to rounding errors and the use of fixed length floating point numbers. In this paper we prove the existence of polynomial time algorithms to approximate the Julia sets of given hyperbolic rational functions. We will give a strict computable error estimation w.r.t. the Hausdorff metric on the complex sphere. This extends a result on polynomials z ↦ → z 2 + c, where |c | < 1/4, in [RW03] and an earlier result in [Zho98] on the recursiveness of the Julia sets of hyperbolic polynomials. The algorithm given in this paper computes Julia sets locally in time O(k · M(k)) (where M(k) denotes the time needed to multiply two k-bit numbers). Roughly speaking, the local time complexity is the number of Turing machine steps to decide a set of disks of spherical diameter 2 −k so that the union of these disks has Hausdorff distance at most 2 −k+2. This allows to give reliable pictures of Julia sets to arbitrary precision. Key words: Julia Sets, Computational Complexity. 1

Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 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

We investigate different definitions of the computability and complexity of sets in R k, and establish new connections between these definitions. This allows us to connect the computability of real functions and real sets in a new way. We show that equivalence of some of the definitions corresponds to equivalence between famous complexity classes. The model we use is mostly consistent with [Wei00]. We apply the concepts developed to show that hyperbolic Julia sets are polynomial time computable. This result is a significant generalization of the result in [RW03], where polynomial time computability has been shown for a restricted type of hyperbolic Julia sets. ii Acknowledgements First of all, I would like to thank my graduate supervisor, Stephen Cook. Our weekly meetings not only allowed me to complete this thesis, but also gave me a much broader and deeper understanding of the entire field of theoretical computer science. Working with him has made my learning process a pleasant one.

We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two–sided distance function of the Mandelbrot set is computable if the hyperbolicity conjecture is true. We formulate the question whether the distance function of the Mandelbrot set is computable also in terms of the escape time.

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.

Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no well-understood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?

... 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.