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Noncomputable Julia sets
 Journ. Amer. Math. Soc
"... 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 computergenerated images of such sets. The algorithms used to draw these pictures vary; ..."
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Cited by 35 (10 self)
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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 computergenerated 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 distanceestimator algorithm [Mil] uses classical complex analysis to give a onepixel 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
On the complexity of real functions
, 2005
"... We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale 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 ..."
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Cited by 18 (5 self)
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We establish a new connection between the two most common traditions in the theory of real computation, the BlumShubSmale 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
Parabolic Julia Sets are Polynomial Time Computable. eprint math.DS/0505036
"... In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time. 1 ..."
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Cited by 16 (5 self)
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In this paper we prove that parabolic Julia sets of rational functions are locally computable in polynomial time. 1
Computability of Julia Sets
, 2008
"... 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. ..."
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Cited by 15 (4 self)
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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.
On computational complexity of Siegel Julia sets
 Commun. Math. Physics
"... 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 ..."
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Cited by 13 (6 self)
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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.
The Mathematician's Bias  and the Return to Embodied Computation. In H. Zenil (Ed.) A Computable Universe. Understanding Computation & Exploring Nature As Computation. World Scientific: New York/London/Singapore
, 2012
"... There are growing uncertainties surrounding the classical model of computation established by Gödel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those more practically engaged in computing, has parallels with th ..."
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Cited by 6 (0 self)
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There are growing uncertainties surrounding the classical model of computation established by Gödel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those more practically engaged in computing, has parallels with the wider one between science and those working creatively or intuitively out in the ‘real ’ world. The scientific outlook is more flexible and basic than some understand or want to admit. The science is subject to limitations which threaten careers. We look at embodiment and disembodiment of computation as the key to the mismatch, and find Turing had the right idea all along – amongst a productive confusion of ideas about computation in the real and the abstract worlds. When we get out of bed in the morning, we approach a complicated world of information with a determination not just to survive the day – though that may be hard enough: we mean to “compute ” our way towards various vaguely defined
Constructing NonComputable Julia Sets
 Proc. of STOC 2007
"... While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unk ..."
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While most polynomial Julia sets are computable, it has been recently shown [12] that there exist noncomputable Julia sets. The proof was nonconstructive, and indeed there were doubts as to whether specific examples of parameters with noncomputable Julia sets could be constructed. It was also unknown whether the noncomputability 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 noncomputable 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 polytime algorithm for computing a number c such that the Julia set J z 2 +cz is noncomputable. In contrast with these results, we show that the filled Julia set of a polynomial is always computable.
CONSTRUCTING LOCALLY CONNECTED NONCOMPUTABLE JULIA SETS
, 2008
"... 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 s ..."
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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 noncomputable, 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.