Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Aspenberg, The Collet-Eckmann condition for rational functions on the Riemann sphere, Math. Z., 273 (2013), 935–980.
A. Avila and C. G. Moreira, Statistical properties of unimodal maps: The quadratic family, Ann. of Math. (2), 161 (2005), 831–881.
I. Binder, M. Braverman and M. Yampolsky, On computational complexity of Siegel Julia sets, Commun. Math. Phys., 264 (2006), 317–334.
I. Binder, M. Braverman and M. Yampolsky, Filled Julia sets with empty interior are computable, Found. Comput. Math., 7 (2007), 405–416.
M. Braverman and M. Yampolsky, Constructing locally connected non-computable Julia sets, Commun. Math. Phys., 291 (2009), 513–532.
M. Braverman, Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are Poly-Time Computable, Master’s thesis, University of Toronto, 2004.
M. Braverman, Parabolic Julia sets are polynomial time computable, Nonlinearity, 19 (2006), 1383–1401.
M. Braverman and M. Yampolsky, Non-computable Julia sets, J. Amer. Math. Soc., 19 (2006), 551–578.
M. Braverman and M. Yampolsky, Computability of Julia Sets, Algorithms and Computation in Mathematics, 23, Springer-Verlag, Berlin, 2009.
J. B. Conway, Functions of One Complex Variable. II, Graduate Texts in Mathematics, 159, Springer-Verlag, New York, 1995.
M. Denker, F. Przytycki and F. Urbański, On the transfer operator for rational functions on the Riemann sphere, Erg. Th. and Dynam Sys. 16 (1996), 255–266.
A. Dudko, Computability of the Julia set. Nonrecurrent critical orbits, Discrete and Continuous Dynamical Systems, Volume 34, 7 (2014), 2751–2778.
A. Dudko, M. Yampolsky, Poly-time computability of Feigenbaum Julia set, Erg. Th. and Dynam. Sys., 36 (2016), 2441–2462
J. Milnor, Self-similarity and hairiness in the Mandelbrot set, “Computers in Geometry and Topology”, Lect. Notes Pure Appl. Math., ed. Tangora, M, 114 (1989), 211–257.
Przytycki, F., Lyapunov Characteristic Exponents are Nonnegative, Proc. Amer. Math. Soc., 119 (1993), No. 1, pp. 309–317
F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math. 155 (1998), no. 2, 189–199.
F. Przytycki, J. Rivera-Letelier, S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math. 151 (2003), 29–63.
R. Rettinger, A Fast Algorithm for Julia Sets of Hyperbolic Rational Functions., Electr. Notes Theor. Comput. Sci., 120 (2005), 145–157.
A. M. Turing, On Computable Numbers, With an Application to the Entscheidungsproblem, Proc. London Math. Soc., 42 (1936), no. 1, 230–265.
H. Weyl, Randbemerkungen zu Hauptproblemen der Mathematik, Math. Z., 20 (1924), 131–150.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stephen Cook.
Michael Yampolsky was partially supported by NSERC Discovery Grant.
Rights and permissions
About this article
Cite this article
Dudko, A., Yampolsky, M. Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set. Found Comput Math 18, 1233–1243 (2018). https://doi.org/10.1007/s10208-017-9367-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10208-017-9367-7