# American Institute of Mathematical Sciences

July  2014, 34(7): 2751-2778. doi: 10.3934/dcds.2014.34.2751

## Computability of the Julia set. Nonrecurrent critical orbits

 1 Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, 11794-3660, United States

Received  June 2012 Revised  September 2013 Published  December 2013

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.
Citation: Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
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##### References:
 [1] Ronnie Pavlov, Pascal Vanier. The relationship between word complexity and computational complexity in subshifts. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1627-1648. doi: 10.3934/dcds.2020334 [2] François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262 [3] Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure & Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211 [4] Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975 [5] Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327 [6] Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583 [7] Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173 [8] Andrea Bonito, Roland Glowinski. On the nodal set of the eigenfunctions of the Laplace-Beltrami operator for bounded surfaces in $R^3$: A computational approach. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2115-2126. doi: 10.3934/cpaa.2014.13.2115 [9] Silvère Gangloff, Benjamin Hellouin de Menibus. Effect of quantified irreducibility on the computability of subshift entropy. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1975-2000. doi: 10.3934/dcds.2019083 [10] Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas. Dynamics and abstract computability: Computing invariant measures. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 193-212. doi: 10.3934/dcds.2011.29.193 [11] Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175 [12] Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929 [13] Silvére Gangloff, Alonso Herrera, Cristobal Rojas, Mathieu Sablik. Computability of topological entropy: From general systems to transformations on Cantor sets and the interval. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4259-4286. doi: 10.3934/dcds.2020180 [14] Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477 [15] Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299 [16] Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167 [17] Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103 [18] Peter Giesl, Sigurdur Hafstein. Computational methods for Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : i-ii. doi: 10.3934/dcdsb.2015.20.8i [19] Bernd Krauskopf, Carlo R. Laing. Novel computational approaches and their applications. Journal of Computational Dynamics, 2020, 7 (2) : i-i. doi: 10.3934/jcd.2020007 [20] Shu Liao, Jin Wang, Jianjun Paul Tian. A computational study of avian influenza. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1499-1509. doi: 10.3934/dcdss.2011.4.1499

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