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Real quadratic Julia sets can have arbitrarily high complexity [article]

Cristobal Rojas, Michael Yampolsky
<span title="2020-03-18">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that there exist real parameters c for which the Julia set J_c of the quadratic map z^2+c has arbitrarily high computational complexity.  ...  This is the first known class of real parameters with a non poly-time computable Julia set.  ...  There exists real parameters c ∈ (−1.75, 0) whose quadratic Julia sets have arbitrarily high computational complexity. Preliminaries Computational Complexity of sets.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.06204v3">arXiv:1904.06204v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ymdm7t5op5hblf7k24enff7bca">fatcat:ymdm7t5op5hblf7k24enff7bca</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200325035043/https://arxiv.org/pdf/1904.06204v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1904.06204v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

On Computational Complexity of Siegel Julia Sets

I. Binder, M. Braverman, M. Yampolsky
<span title="2006-03-22">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/4t2npfwxdfchbkc6v2dtzlglle" style="color: black;">Communications in Mathematical Physics</a> </i> &nbsp;
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.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00220-006-1546-3">doi:10.1007/s00220-006-1546-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bn7cnpo5u5d3baizv55qhcr36m">fatcat:bn7cnpo5u5d3baizv55qhcr36m</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190308094531/http://pdfs.semanticscholar.org/e655/9dd601dc4dc8fb2a49ab1fde58982cb77c80.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e6/55/e6559dd601dc4dc8fb2a49ab1fde58982cb77c80.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00220-006-1546-3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Constructing non-computable Julia sets

Mark Braverman, Michael Yampolsky
<span title="">2007</span> <i title="ACM Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jlc5kugafjg4dl7ozagimqfcbm" style="color: black;">Proceedings of the thirty-ninth annual ACM symposium on Theory of computing - STOC &#39;07</a> </i> &nbsp;
In fact, in the case of Julia sets of quadratic polynomials we give a precise characterization of Julia sets with computable parameters.  ...  It was also unknown whether the non-computability proof can be extended to the filled Julia sets.  ...  Acknowledgments We would like to thank John Milnor for posing the question of computability of filled Julia sets to us.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1145/1250790.1250893">doi:10.1145/1250790.1250893</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/stoc/BravermanY07.html">dblp:conf/stoc/BravermanY07</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/s5xwtsfxbfhhvmxyxquexhljzi">fatcat:s5xwtsfxbfhhvmxyxquexhljzi</a> </span>
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Almost every real quadratic polynomial has a poly-time computable Julia set [article]

Artem Dudko, Michael Yampolsky
<span title="2017-08-09">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that Collet-Eckmann rational maps have poly-time computable Julia sets. As a consequence, almost all real quadratic Julia sets are poly-time.  ...  Indeed, in [3] it was shown that there exist computable quadratic Julia sets with an arbitrarily high time complexity.  ...  However, even a computable Julia set could have such a high computational complexity as to render any practical simulations impossible.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.05768v2">arXiv:1702.05768v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yggkxuwztjefjdfkxwt5ozlycq">fatcat:yggkxuwztjefjdfkxwt5ozlycq</a> </span>
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On computational complexity of Cremer Julia sets

Artem Dudko, Michael Yampolsky
<span title="">2020</span> <i title="Institute of Mathematics, Polish Academy of Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yeobluctzze7rfwekxs7sx2ayy" style="color: black;">Fundamenta Mathematicae</a> </i> &nbsp;
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.  ...  In §2 we introduce the main tools of complex dynamics used in the proof. We present the proof in §3. 1.1. Cremer quadratic Julia sets.  ...  The Fatou-Shishikura bound implies that a quadratic polynomial can have at most one periodic point whose multiplier λ is in {|z| ≤ 1}.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4064/fm829-12-2019">doi:10.4064/fm829-12-2019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6upudnvulfdhjdqztzjoz6jhca">fatcat:6upudnvulfdhjdqztzjoz6jhca</a> </span>
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On computational complexity of Cremer Julia sets [article]

Artem Dudko, Michael Yampolsky
<span title="2019-10-07">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We find an abundance of Cremer Julia sets of an arbitrarily high computational complexity.  ...  (II) Does there exist at least one Cremer Julia set for which every algorithm will have an impractical running time (i.e. can we prove that there is at least one such J c with a high, for instance, non-polynomial  ...  Fatou-Shishikura bound implies that a quadratic polynomial p c can have at most one periodic point whose multiplier λ ∈ {|z| ≤ 1}.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1907.11047v3">arXiv:1907.11047v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rcl5rd72pbfyboguyors4vqbam">fatcat:rcl5rd72pbfyboguyors4vqbam</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200824191104/https://arxiv.org/pdf/1907.11047v3.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/31/8a/318a4cc52fed15bc63171ffe17367b7da619b953.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1907.11047v3" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize

Mikhail Lyubich
<span title="">2012</span> <i title="American Institute of Mathematical Sciences (AIMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ebeyl2om5vbsrbiuzncpyjustm" style="color: black;">Journal of Modern Dynamics</a> </i> &nbsp;
(Note that the Julia set J ( f c ) of a quadratic polynomial coincides with the Julia set of its quadratic-like restriction to a sufficiently large disk.) 1 The standard parametrization of the complex  ...  It is shown in [1] that in fact, the Hausdorff dimension of Feigenbaum Julia sets can be arbitrarily close to 1. ( f k x) → φ d µ for any φ ∈ C (I ) and a.e. x ∈ I . 1. 7 . 7 Milnor's problem.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.3934/jmd.2012.6.183">doi:10.3934/jmd.2012.6.183</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/he35vwtpkfeulc3jiutv5lnuya">fatcat:he35vwtpkfeulc3jiutv5lnuya</a> </span>
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Fractal Geography of the Riemann Zeta Function [article]

Chris King
<span title="2011-09-23">2011</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The quadratic Mandelbrot set has been referred to as the most complex and beautiful object in mathematics and the Riemann Zeta function takes the prize for the most complicated and enigmatic function.  ...  Here we elucidate the spectrum of Mandelbrot and Julia sets of Zeta, to unearth the geography of its chaotic and fractal diversities, combining these two extremes into one intrepid journey into the deepest  ...  Evidence of this can also be seen in the Julia sets, by comparison with a base Julia set for the quadratic satellite.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1103.5274v3">arXiv:1103.5274v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e4al4rj7rvbbxo47saacdlul2m">fatcat:e4al4rj7rvbbxo47saacdlul2m</a> </span>
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Filled Julia sets with empty interior are computable [article]

I. Binder, M. Braverman, M. Yampolsky
<span title="2006-06-27">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that if a polynomial filled Julia set has empty interior, then it is computable.  ...  The machine computing J R is given access to R(z) through an oracle that can provide an approximation of any coefficient of R(z) with an arbitrarily high (but finite) requested precision.  ...  Given a compact set S ⊂ R k , our goal is to be able to approximate the set S with an arbitrarily high precision. Here we ought to specify what do "approximate" and "precision" mean in this setting.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0410580v3">arXiv:math/0410580v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/czsrnlofrrafdagiwkhswlgvee">fatcat:czsrnlofrrafdagiwkhswlgvee</a> </span>
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Ergodicity of conformal measures for unimodal polynomials [article]

Eduardo A. Prado
<span title="1996-06-15">1996</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We show that for any unimodal polynomial f with real coefficients, all conformal measures for f are ergodic.  ...  A Lyubich polynomial is an infinitely many times renormalizable quadratic polynomial in SL with sufficiently high combinatorics as described in [Lyu93] .  ...  map with connected filled in Julia set.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/9606222v1">arXiv:math/9606222v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/e3w2ywh72rg3rbifq332z7nlii">fatcat:e3w2ywh72rg3rbifq332z7nlii</a> </span>
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Computable geometric complex analysis and complex dynamics [article]

Cristobal Rojas, Michael Yampolsky
<span title="2017-03-19">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As applications, we review the state of the art regarding computability and complexity of Julia sets, their invariant measures and external rays impressions.  ...  We discuss computability and computational complexity of conformal mappings and their boundary extensions.  ...  There exist Siegel quadratics of the form P θ (z) = e 2πiθ z+z 2 whose Julia set have an arbitrarily high time complexity. Given a lower complexity bound, such a θ can be produced constructively.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1703.06459v1">arXiv:1703.06459v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qxmigwp56fcmrpha4ubkvunxii">fatcat:qxmigwp56fcmrpha4ubkvunxii</a> </span>
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On computability of Julia sets: answers to questions of Milnor and Shub [article]

Mark Braverman, Michael Yampolsky
<span title="2006-04-07">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ...  This implies, in particular, that all Cremer quadratic Julia sets are computablethis despite the fact that no informative high resolution images of such sets have ever been produced.  ...  One expects, however, that such "bad" but still computable examples have high algorithmic complexity, which makes the computational cost of producing such a picture prohibitively high.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0604175v1">arXiv:math/0604175v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/tijjyywdafhn3gwb7kenw7tjpa">fatcat:tijjyywdafhn3gwb7kenw7tjpa</a> </span>
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Filled Julia Sets with Empty Interior Are Computable

I. Binder, M. Braverman, M. Yampolsky
<span title="2006-12-28">2006</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2joypbq7ubehha6yu7ylkxkoqa" style="color: black;">Foundations of Computational Mathematics</a> </i> &nbsp;
We show that if a polynomial filled Julia set has empty interior, then it is computable.  ...  The authors wish to thank John Milnor, whose encouragement and questions have inspired this work.  ...  Introduction We refer the reader to [Mil] for the basic definitions of Complex Dynamics, such as the Julia set J f of a rational map f :Ĉ →Ĉ, and the filled Julia set K p of a polynomial p : C → C.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10208-005-0210-1">doi:10.1007/s10208-005-0210-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cpjdi75rfnhylh7utcrlatwwnu">fatcat:cpjdi75rfnhylh7utcrlatwwnu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190225090138/http://pdfs.semanticscholar.org/5d80/a62b29c9b9522964945e97164e7c1fa619c9.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/5d/80/5d80a62b29c9b9522964945e97164e7c1fa619c9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10208-005-0210-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Poly-time computability of the Feigenbaum Julia set

ARTEM DUDKO, MICHAEL YAMPOLSKY
<span title="2015-07-21">2015</span> <i title="Cambridge University Press (CUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/t7qnh4dicrhptozut7w2ifbxua" style="color: black;">Ergodic Theory and Dynamical Systems</a> </i> &nbsp;
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.  ...  Such parameters are rare, however; for almost every c ∈ C the set J c is computable. In [1] it was shown that there exist computable quadratic Julia sets with an arbitrarily high time complexity.  ...  Is the Julia set of a typical real quadratic map poly-time?  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1017/etds.2015.24">doi:10.1017/etds.2015.24</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nuwn2hl2n5hjbkmdag5jgup5ju">fatcat:nuwn2hl2n5hjbkmdag5jgup5ju</a> </span>
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Computability of Julia sets [article]

Mark Braverman, Michael Yampolsky
<span title="2007-09-29">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
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.  ...  Binder, we have shown that computational complexity of quadratic Julia sets can be arbitrarily high.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0610340v2">arXiv:math/0610340v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mykgrczqpzgvzo2t3b2wtcxpr4">fatcat:mykgrczqpzgvzo2t3b2wtcxpr4</a> </span>
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