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Computability of Julia sets [article]

Mark Braverman, Michael Yampolsky
2007 arXiv   pre-print
We also show that a filled Julia set of a polynomial is always computable.  ...  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.  ...  Lack of uniform computability of Julia sets.  ... 
arXiv:math/0610340v2 fatcat:mykgrczqpzgvzo2t3b2wtcxpr4

Computability of Julia Sets

M. Braverman, M. Yampolsky
2008 Moscow Mathematical Journal  
We also show that a filled Julia set of a polynomial is always computable.  ...  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 thank John Milnor for posing the question on computability of filled Julia sets to us.  ... 
doi:10.17323/1609-4514-2008-8-2-185-231 fatcat:x6ymucv2m5dbxdqdak5v3snppi

Computability and Complexity of Julia Sets

Kota Hiratsuka, Yuzuru Sato, Zin Arai
2014 IEICE Proceeding Series  
In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3].  ...  Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1][10][13].  ...  Tomoo Yokoyama (Kyoto University of Education) for useful discussion.  ... 
doi:10.15248/proc.2.2 fatcat:ewbeetz6bffopnmxduqxfh26qq

On computational complexity of Cremer Julia sets

Artem Dudko, Michael Yampolsky
2020 Fundamenta Mathematicae  
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.  ...  Where it does not lead to confusion, we will write J θ , K θ , etc., for the Julia set and the filled Julia set of f θ .  ...  The structure of the paper is as follows. In §1.1 we briefly review the definitions of Cremer Julia sets.  ... 
doi:10.4064/fm829-12-2019 fatcat:6upudnvulfdhjdqztzjoz6jhca

On computational complexity of Cremer Julia sets [article]

Artem Dudko, Michael Yampolsky
2019 arXiv   pre-print
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.  ...  Where it does not lead to a confusion, we will write J θ , K θ , etc., for the Julia set and the filled Julia set of f θ .  ...  (I) Does there exist an algorithm to compute at least one Cremer Julia sets with a practical running time (for instance, polynomial in n, where 2 −n is the size of the pixel on the computer screen)?  ... 
arXiv:1907.11047v3 fatcat:rcl5rd72pbfyboguyors4vqbam

On Computational Complexity of Siegel Julia Sets

I. Binder, M. Braverman, M. Yampolsky
2006 Communications in Mathematical Physics  
In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high.  ...  It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification.  ...  There exist quadratic Siegel Julia sets of arbitrarily high computational complexity.  ... 
doi:10.1007/s00220-006-1546-3 fatcat:bn7cnpo5u5d3baizv55qhcr36m

The Computational Complexity of Some Julia Sets

Robert Rettinger, Klaus Weihrauch
2002 Electronical Notes in Theoretical Computer Science  
Numerous computer programs have been written to compute sets of points which approximate Julia sets [4] .  ...  Algorithms for computing approximations of Julia sets can be found e.g. in [9, 7] . In [11] it is shown that many Julia sets are recursive.  ...  Then the Julia set J of the function z 2 + c can be computed locally in time O(k 2 · M (k)).  ... 
doi:10.1016/s1571-0661(04)80386-4 fatcat:va3gtnjwhbejbf6ozivymts554

Computability of the Julia set. Nonrecurrent critical orbits [article]

Artem Dudko
2011 arXiv   pre-print
We prove that the Julia set of a rational function f is computable in polynomial time, assuming that the postcritical set of f does not contain any critical points or parabolic periodic orbits.  ...  Also, there are examples of computable Julia sets whose computational complexity is arbitrarily high (see [3] ).  ...  One of the natural open questions of computational complexity of Julia sets is how large is the class of rational functions (in a sense of Lebesgue measure on the parameter space) whose Julia set can be  ... 
arXiv:1109.2946v2 fatcat:z6ajm6lchvb4rc6vvcn7wwnwg4

Computability and complexity of Julia sets: a review

Kota Hiratsuka, Yuzuru Sato, Zin Arai
2014 Nonlinear Theory and Its Applications IEICE  
In this review paper, we introduce a recently developed computability theory for Julia sets in complex dynamical systems by Braverman and Yampolsky [3].  ...  Turing introduced the notion of computability in 1936, various theories of real number computation have been studied [1, 10, 13].  ...  Tomoo Yokoyama (Kyoto University of Education) for useful discussion.  ... 
doi:10.1587/nolta.5.410 fatcat:rjexh5kffngsbjbfs62w2hwwti

Poly-time computability of the Feigenbaum Julia set

ARTEM DUDKO, MICHAEL YAMPOLSKY
2015 Ergodic Theory and Dynamical Systems  
We present the first example of a poly-time computable Julia set with a recurrent critical point: we prove that the Julia set of the Feigenbaum map is computable in polynomial time.  ...  As follows from [2] , the Julia set of f c * is computable.  ...  Hence, our main result -poly-time computability of the Julia set of F -can be stated without the use of an oracle for the map F .  ... 
doi:10.1017/etds.2015.24 fatcat:nuwn2hl2n5hjbkmdag5jgup5ju

Computer Visualization of Julia Sets for Maps beyond Complex Analyticity

Toporensky Alexey, Stepanyan Ivan
2021 EPJ Web of Conferences  
Using the computer program creating Julia sets for two-dimensional maps we have made computer animation showing how Julia sets for the Peckham map alters when the parameter of the map is changing.  ...  Computer animation of Julia fractal sets allows seeing how patterns typical for complex maps gradually destroy.  ...  set of straight lines and the pa per [ 6] which de scribes t he c onditions of existence of Sierpiński carpet Julia sets and estimation of it's Hausdorff dimension.  ... 
doi:10.1051/epjconf/202124801013 doaj:917291b8c0354f3eaee2527399b7b848 fatcat:7kszefswmzhblpaphzwnsah2b4

On computability of Julia sets: answers to questions of Milnor and Shub [article]

Mark Braverman, Michael Yampolsky
2006 arXiv   pre-print
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.  ...  Two questions on computability of Julia sets. J. Milnor has asked us the following natural question: Is the filled Julia set of a quadratic polynomial always computable?  ...  Introduction Computability of real sets. The reader is directed to [BY] for a more detailed discussion of the notion of computability of subsets of R n as applied, in particular, to Julia sets.  ... 
arXiv:math/0604175v1 fatcat:tijjyywdafhn3gwb7kenw7tjpa

A Multidimensional Artistic Approach to Enhance Understanding of Julia Sets through Computer Programming

E. A. Navas-López
2022 International Journal on Integrating Technology in Education  
This article proposes an artistic approach to increase and enrich the understanding of Julia Sets. This approach includes the mathematical, the playful, the artistic and the computational dimensions.  ...  Four artistic creations based on Julia Sets are presented as examples using the CFDG language.  ...  Section 2 presents four examples of computer-generated artworks based on Julia sets to use as a starting point for the proposal.  ... 
doi:10.5121/ijite.2022.11103 fatcat:7u6n6s7bkjca3mg2mt63vewfva

Construction of 3D Mandelbrot Set and Julia Set

Ashish Negi, Ankit Garg, Akshat Agrawal
2014 International Journal of Computer Applications  
This paper present construction of famous fractal images-Mandelbrot set and Julia set using 3D iterated function system which gives real look and feel of complex natural fractal images.  ...  Today Fractal geometry is completely new area of research in the field of computer science and engineering. It has wide range of applications.  ...  Asish Negi, Associate professor in Department of Computer Science and Engineering, Govind Ballabh pant Engineering College, Pauri Garhwal for providing me an opportunity and resource to carry out this  ... 
doi:10.5120/14920-3514 fatcat:nubtgttyqrbxvewl2tshaoldku

Julia Sets and Mandelbrot-Like Sets Associated With Higher Order Schroder Rational Iteration Functions: A Computer Assisted Study

Edward R. Vrscay
1986 Mathematics of Computation  
The Julia sets for the Sm(z), as applied to the simple cases g"(z) = z" -1, are examined for increasing m with the help of microcomputer plots.  ...  For g(z) a polynomial, this involves the iteration of rational functions over the complex Riemann sphere, which is described by the classical theory of Julia and Fatou and subsequent developments.  ...  Julia sets of rational maps over the complex sphere.  ... 
doi:10.2307/2008220 fatcat:fp7eaasx3vgvtang6qwq4d5iyu
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