## Filled Julia sets with empty interior are computable. e-print, math.DS/0410580

Citations: | 14 - 8 self |

### BibTeX

@MISC{Binder_filledjulia,

author = {I. Binder and M. Braverman and M. Yampolsky},

title = {Filled Julia sets with empty interior are computable. e-print, math.DS/0410580},

year = {}

}

### OpenURL

### Abstract

Abstract. We show that if a polynomial filled Julia set has empty interior, then it is computable. 1.

### Citations

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Computable Analysis
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Citation Context ...e analysis. Their roots can be traced to the pioneering work of Banach and Mazur of 1937 (see [Maz]). The reader may find an extended exposition of the model of computation we are using in [BrC]. See =-=[Wei]-=- for a detailed discussion of the concepts of modern computable analysis. See also [BY] for a discussion of computability as applied to problems in Complex Dynamics. Given a compact set S ⊂ Rk , our g... |

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Citation Context ... elementary consequence of Koebe Theorem and the constant may be chosen as C2 = 4 � r(U, u) (see [RZ], for example). The other inequality may be found in many advanced texts on Complex Analysis, e.g. =-=[Pom]-=-. Proof of Proposition 3.4. We will show the computability of the inner radius of ∆θ. The algorithm works as follows: (I) For k ∈ N compute a set Dk ∈ C which is a 2 −m -approximation of the preimage ... |

154 |
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Citation Context ...is computable. 1. Introduction Julia sets of rational mappings. We recall the main definitions of complex dynamics relevant to our results only briefly; a good general reference is the book of Milnor =-=[Mil]-=-. For a rational mapping R of degree deg R = d ≥ 2 considered as a dynamical system on the Riemann sphere R : Ĉ → Ĉ the Julia set is defined as the complement of the set where the dynamics is Lyapunov... |

36 |
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Citation Context ... d rational maps without rotation domains or parabolic orbits. Then a Julia set depends continuously, in Hausdorff sense on Ĉ, on the parameter in Rd . While this is well-known (see the discussion in =-=[Do]-=-), it is notable that the proof of this dynamical statement is produced by a computer science argument. 1 Acknowledgement. The authors wish to thank John Milnor, whose encouragement and questions have... |

31 | Self-similarity of Siegel disks and Hausdorff dimension of Julia sets
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Citation Context ... golden mean, are indeed computable by this recipe: Proposition 3.2. The inner radius ρθ∗ is a computable real number. Proof. We appeal to the renormalization theory for golden-mean Siegel disks (see =-=[McM]-=-), which implies, in particular, that the boundary of ∆θ∗ is self-similar up to an exponentially small error. As a consequence, there exist C > 0, and λ > 1 such that inf{|f i (0)|, i = 0, . . ., qn} ... |

27 | Yampolski M.: Non-computable Julia sets
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Citation Context ... (see [Maz]). The reader may find an extended exposition of the model of computation we are using in [BrC]. See [Wei] for a detailed discussion of the concepts of modern computable analysis. See also =-=[BY]-=- for a discussion of computability as applied to problems in Complex Dynamics. Given a compact set S ⊂ Rk , our goal is to be able to approximate the set S with an arbitrarily high precision. Here we ... |

23 | The Computational Complexity of Some Julia Sets - Rettinger, Weihrauch |

20 |
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Citation Context ... note that for any polynomial Q(z), we can list rational approximations r1, r2, . . .,rm of all the roots α1, α2, . . .,αm of Q(z) − z with an arbitrarily good precision 2−k (a classical reference is =-=[Wey]-=-). Let M > 0 be some bound on |Q ′′ (z)| in the area of the roots. Then |Q ′ (ri)| > 1+2 −kM implies that |Q ′ (αi)| > 1, and in fact acts as a certificate that this is the case. It is easy to see tha... |

16 | A fast algorithm for Julia sets of hyperbolic rational functions
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Citation Context ...= Kp). Then Kp is computable by M φ . The known results in this direction, which make the statement of the theorem interesting, are the following. Independently, the second author [Brv] and Rettinger =-=[Ret]-=- have demonstrated that hyperbolic Julia sets are polynomial-time computable by an oracle TM. In sharp contrast, in [BY] the second and third authors demonstrated the existence of non-computable Julia... |

15 |
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Citation Context ...luate the conformal radius r(Dk, 0) with precision 2 −(m+1) (this can be done, for example, by using one of the numerous existing methods for computing the Riemann Mapping of a computable domain, see =-=[Zho]-=-, for example), if necessairy, compute Dk or parts of Dk with a higher degree of precision;s10 I. BINDER, M. BRAVERMAN, M. YAMPOLSKY (III) if r(Dk, 0) is 2 −m C1-close to rθ then compute the inner rad... |

12 | Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are PolyTime Computable
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(Show Context)
Citation Context ...nterior (so that Jp = Kp). Then Kp is computable by M φ . The known results in this direction, which make the statement of the theorem interesting, are the following. Independently, the second author =-=[Brv]-=- and Rettinger [Ret] have demonstrated that hyperbolic Julia sets are polynomial-time computable by an oracle TM. In sharp contrast, in [BY] the second and third authors demonstrated the existence of ... |

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7 |
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Citation Context ...c polynomials. The latter examples are given by polynomials with Siegel disks. It is shown in [BY] that there exist non-computable examples with rather wild topology. By a method of Buff and Chéritat =-=[BC]-=- the boundary of the Siegel disk can be made smooth, and as a consequence, the critical point is not in Date: March 12, 2008. The first and third authors are partially supported by NSERC Discovery gra... |

5 | Variation of the conformal radius
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Citation Context ... and C2 = C2(r(U, u)) such that √ C1|ρ − ρ1| ≤ |r(U, u) − r(U1, u)| ≤ C2 ǫ. The second inequality is an elementary consequence of Koebe Theorem and the constant may be chosen as C2 = 4 � r(U, u) (see =-=[RZ]-=-, for example). The other inequality may be found in many advanced texts on Complex Analysis, e.g. [Pom]. Proof of Proposition 3.4. We will show the computability of the inner radius of ∆θ. The algori... |

3 |
Computable analysis, Rosprawy Matematyczne
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Citation Context ...ets of Rn . The computability notions that we apply for subsets of Rk belong to the framework of computable analysis. Their roots can be traced to the pioneering work of Banach and Mazur of 1937 (see =-=[Maz]-=-). The reader may find an extended exposition of the model of computation we are using in [BrC]. See [Wei] for a detailed discussion of the concepts of modern computable analysis. See also [BY] for a ... |

2 |
Computing over the Reals
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Citation Context ...f computable analysis. Their roots can be traced to the pioneering work of Banach and Mazur of 1937 (see [Maz]). The reader may find an extended exposition of the model of computation we are using in =-=[BrC]-=-. See [Wei] for a detailed discussion of the concepts of modern computable analysis. See also [BY] for a discussion of computability as applied to problems in Complex Dynamics. Given a compact set S ⊂... |

2 |
Recent results on the boundaries of Siegel disks. Progress in holomorphic dynamics, 41–49, Pitman Res
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Citation Context ... a method of Buff and Chéritat [BC] the boundary of the Siegel disk can be made smooth, and as a consequence, the critical point is not in the boundary and the Julia set is not locally connected (see =-=[Ro]-=- for a discussion of the topological anomalies in such Julia sets). As our present theorem shows, however, the notions of topological complexity and computational complexity are rather distinct. It co... |