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Almost every real quadratic polynomial has a poly-time computable Julia set [article]

Artem Dudko, Michael Yampolsky
2017 arXiv   pre-print
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.  ...  Our main result gives a positive answer for Question 2 in the case of real parameters c: Main Theorem. For almost every real value of the parameter c, the Julia set J c is poly-time.  ...  Poly-time computability has been previously established for several types of quadratic Julia sets. Firstly, all hyperbolic Julia sets are poly-time [6, 18] .  ... 
arXiv:1702.05768v2 fatcat:yggkxuwztjefjdfkxwt5ozlycq

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

Cristobal Rojas, Michael Yampolsky
2020 arXiv   pre-print
This is the first known class of real parameters with a non poly-time computable Julia set.  ...  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.  ...  In this case, it was recently proved by Dudko and Yampolsky [DY17] that almost every real quadratic Julia set is poly-time computable.  ... 
arXiv:1904.06204v3 fatcat:ymdm7t5op5hblf7k24enff7bca

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.  ...  The second question has already been formulated in [11] : Question. Is the Julia set of a typical real quadratic map poly-time?  ...  For every infinitely renormalizable real quadratic polynomial f c its Julia set is polytime computable with an oracle for c.  ... 
doi:10.1017/etds.2015.24 fatcat:nuwn2hl2n5hjbkmdag5jgup5ju

Computable geometric complex analysis and complex dynamics [article]

Cristobal Rojas, Michael Yampolsky
2017 arXiv   pre-print
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.  ...  Almost every real quadratic Julia set is poly-time. This means that poly-time computability is a "physically natural" property in real dynamics.  ... 
arXiv:1703.06459v1 fatcat:qxmigwp56fcmrpha4ubkvunxii

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.  ...  Julia sets are poly-time computable. It is an open question whether any polynomial Julia set with a parabolic point is not poly-time. The structure of the paper is as follows.  ... 
doi:10.1007/s00220-006-1546-3 fatcat:bn7cnpo5u5d3baizv55qhcr36m

Basins of Attraction for Various Steffensen-Type Methods

Alicia Cordero, Fazlollah Soleymani, Juan R. Torregrosa, Stanford Shateyi
2014 Journal of Applied Mathematics  
The computer programming package MATHEMATICAprovides a powerful but easy environment for all aspects of numerics.  ...  This paper puts on show one of the application of this computer algebra system in finding fixed points of iteration functions.  ...  To be more precise (see [31, 32] ), a given method is generally convergent if the scheme converges to a root for almost every starting point and for almost every polynomial of a given degree.  ... 
doi:10.1155/2014/539707 fatcat:iyaqqnzwv5f4ljaeqoj4rust4a

Julia sets for the super-Newton method, Cauchy's method, and Halley's method

Kyle Kneisl
2001 Chaos  
Using the concept of a universal Julia set ͑motivated by the results of McMullen͒, we establish that these algorithms converge when applied to any quadratic with distinct roots.  ...  We include computer plots showing the dynamic structure for each algorithm applied to a variety of polynomials.  ...  ACKNOWLEDGMENTS This work constitutes a portion of the author's dissertation research presently underway under the guidance of Jane Hawkins at the University of North Carolina at Chapel Hill.  ... 
doi:10.1063/1.1368137 pmid:12779470 fatcat:5dfgmyr7trekxes4d7ahfrylwe

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.  ...  Then there is a poly-time computable θ (and hence a poly-time computable c = c(θ)) such that r(θ) = r. Proof.  ... 
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 are grateful to Michael Shub for formulating a question which has motivated much of the discussion on the "shape" of uncomputable Julia sets in this paper.  ... 
doi:10.17323/1609-4514-2008-8-2-185-231 fatcat:x6ymucv2m5dbxdqdak5v3snppi

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.  ...  set is poly-time computable.  ...  Similarly, K is poly-time computable as a subset ofĈ if and only if it is poly-time computable as a subset of R 2 . In this paper we discuss computability of the Julia sets of rational functions.  ... 
arXiv:1109.2946v2 fatcat:z6ajm6lchvb4rc6vvcn7wwnwg4

A survey on real structural complexity theory

Klaus Meer, Christian Michaux
1997 Bulletin of the Belgian Mathematical Society Simon Stevin  
In this tutorial paper we overview research being done in the field of structural complexity and recursion theory over the real numbers and other domains following the approach by Blum, Shub, and Smale  ...  Lickteig for helpful comments on a previous version of this paper.  ...  According to Theorem 4.1 a weak machine solving the problem has to compute a multiple of p in weak polynomial time : this is not possible since the degree of p has to be polynomial in the time of the computation  ... 
doi:10.36045/bbms/1105730626 fatcat:g6j5ofmchbasxotchnf6pgl4gi

Discontinuity of Straightening in Anti-holomorphic Dynamics: I [article]

Hiroyuki Inou, Sabyasachi Mukherjee
2020 arXiv   pre-print
This is the first known example of discontinuity of straightening maps on a real two-dimensional slice of an analytic family of holomorphic polynomials.  ...  We study baby Tricorns appearing in the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials, and show that the dynamically natural straightening map from a baby Tricorn  ...  along a Cantor set of points on the Julia set.  ... 
arXiv:1605.08061v4 fatcat:4qpgl4zy7jephametfm65kq2xe

Quadratic Goldreich-Levin Theorems

Madhur Tulsiani, Julia Wolf
2011 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science  
We give a polynomial time algorithm for computing such a decomposition.  ...  Given a function f : F n 2 → {−1, 1} at fractional Hamming distance 1/2 − ε from a quadratic phase (which is a codeword of Reed-Muller code of order 2), we give an algorithm that runs in time polynomial  ...  They showed that there exists a subspace of codimension poly(1/ε) and on all of whose cosets f correlates polynomially with a quadratic phase.  ... 
doi:10.1109/focs.2011.59 dblp:conf/focs/TulsianiW11 fatcat:pldj2gqt6vgarmiy4gf2bqvqfu

Quadratic Goldreich--Levin Theorems

Madhur Tulsiani, Julia Wolf
2014 SIAM journal on computing (Print)  
We give a polynomial time algorithm for computing such a decomposition.  ...  Given a function f : F n 2 → {−1, 1} at fractional Hamming distance 1/2 − ε from a quadratic phase (which is a codeword of Reed-Muller code of order 2), we give an algorithm that runs in time polynomial  ...  They showed that there exists a subspace of codimension poly(1/ε) and on all of whose cosets f correlates polynomially with a quadratic phase.  ... 
doi:10.1137/12086827x fatcat:lrqygrc6mja7zl6jvxjiajy77q

Quadratic Goldreich-Levin Theorems [article]

Madhur Tulsiani, Julia Wolf
2011 arXiv   pre-print
We give a polynomial time algorithm for computing such a decomposition.  ...  Given a function f:_2^n →{-1,1} at fractional Hamming distance 1/2-ϵ from a quadratic phase (which is a codeword of Reed-Muller code of order 2), we give an algorithm that runs in time polynomial in n  ...  We prove the decomposition theorem using a procedure which, at every step, tests if a certain function has correlation at least 1/2− ε with a quadratic phase.  ... 
arXiv:1105.4372v1 fatcat:dpjgthpi4ja7nfttm62pkduhze
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