## Non-computable Julia sets

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Venue: | Journ. Amer. Math. Soc |

Citations: | 27 - 6 self |

### BibTeX

@ARTICLE{Braverman_non-computablejulia,

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

title = {Non-computable Julia sets},

journal = {Journ. Amer. Math. Soc},

year = {},

volume = {19},

pages = {2006}

}

### OpenURL

### Abstract

Polynomial Julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system. Apart from the beautiful mathematics, one of the reasons for their popularity is the beauty of the computer-generated images of such sets. The algorithms used to draw these pictures vary; the most naïve work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor’s distance-estimator algorithm [Mil] uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution. Moreover, there are examples, even in the family of quadratic polynomials, when no satisfactory pictures of the Julia set exist. In this paper we study computability properties of Julia sets of quadratic polynomials. Under the definition we use, a set is computable, if, roughly speaking, its image can be generated by a computer with an arbitrary precision. Under this notion of computability we show: Main Theorem. There exists a parameter value c ∈ C such that the Julia set of

### Citations

1178 |
On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1937
(Show Context)
Citation Context ... AND M. YAMPOLSKY In the formal setting for the study of computability theory computations are performed by objects called Turing Machines. Turing Machines were introduced in 1936 by Alan Turing (see =-=[Tur]-=-) and are accepted by the scientific community as the standard model of computation. The Turing Machine (TM in short) is capable of solving exactly the same problems as an ordinary computer. Most of t... |

530 |
Introduction to the Theory of Computation
- Sipser
- 1997
(Show Context)
Citation Context ...me, one can think of the TM as a computer program written in any programming language. It is important to mention that there are only countably many TMs, which can be enumerated in a natural way. See =-=[Sip]-=- for a formal discussion on TMs. We define computability as follows. Definition 2.1. We say that a function f : {0, 1} ∗ →{0, 1} ∗ is computable if there is a TM, which on input string s outputs the s... |

409 | A (1967), Foundations of Constructive Analysis
- Bishop
(Show Context)
Citation Context ...⊂ ∆θ0 and dH(∂V,∂∆θ0 ) <ɛ. By Schwarz’s Lemma, the conformal radius r(θ0) < 2. Hence, by Lemma 2.15, |r(V,0) − r(θ0)| <δ=8 √ ɛ. Using any constructive version of the Riemann Mapping Theorem (see e.g. =-=[BB]-=-), we can compute r(V,0) to precision δ and hence know r(θ0) uptoanerrorof 2δ. Given that δ can be made arbitrarily small, we have shown that r(θ0) is computable. � We also state for future reference ... |

237 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...o binary strings. Namely C is the set of finite unions of dyadic balls: � n� � C = B(di,ri) | where di,ri ∈ D . i=1 The following definition is equivalent to the set computability definition given in =-=[Wei]-=- (see also [RW]). Definition 2.3. WesaythatacompactsetK ⊂ R k is computable if there exists a TM M(m), such that on an input m ∈ N,themachineM(m) outputs an encoding of Cm ∈Csuch that dH(K,Cm) < 2 −m ... |

163 |
Complexity theory of real functions
- Ko
- 1991
(Show Context)
Citation Context ...h an access to arbitrarily good approximations for x, M should be able to produce an arbitrarily good approximation for f(x). This definition trivially generalizes to domains of higher dimension. See =-=[Ko1]-=- for more details. One of the most important properties of computable functions is that Proposition 2.1. Computable functions are continuous.sNON-COMPUTABLE JULIA SETS 553 Let K ⊂ R k be a compact set... |

147 |
Hilbert’s Tenth Problem
- Matiyasevich
- 1993
(Show Context)
Citation Context ... many non-computable functions and undecidable sets. The best-known intractable problems are the Halting Problem and the solvability of a Diophantine equation (Hilbert’s 10-th problem); see [Sip] and =-=[Mat]-=- for more information. Computability of real functions and sets. In the present paper we are interested in the computability of functions f : R n → R and subsets of R n ,particularly subsets of R 2 ∼ ... |

146 | Complex dynamics and renormalization - McMullen - 1994 |

141 | On the dynamics of polynomial-like mappings - Douady, Hubbard - 1985 |

136 |
Boundary Behaviour of Conformal Maps
- Pommerenke
- 1991
(Show Context)
Citation Context ...) establishes a bijection between marked topological disks properly contained in C and univalent maps φ : D → C with φ ′ (0) > 0. The following theorem is due to Carathéodory. A proof may be found in =-=[Pom]-=-. Theorem 2.13 (Carathéodory Kernel Theorem). The mapping ι is a homeomorphism with respect to the Carathéodory topology on domains and the compactopen topology on maps. Proposition 2.14. The conforma... |

133 |
dynamics” Dynamics in one complex variable. Introductory lectures
- Milnor
- 1999
(Show Context)
Citation Context ...s of such sets. The algorithms used to draw these pictures vary; the most naïve work by iterating the center of a pixel to determine if it lies in the Julia set. Milnor’s distance-estimator algorithm =-=[Mil]-=- uses classical complex analysis to give a one-pixel estimate of the Julia set. This algorithm and its modifications work quite well for many examples, but it is well known that in some particular cas... |

100 |
Étude dynamique des polynômes complexes I
- Douady, Hubbard
- 1984
(Show Context)
Citation Context ...t parameter values for which fc has a parabolic point. We, however, will employ a more obvious discontinuity which is related to Siegel disks. Let us first note that by a result of Douady and Hubbard =-=[DH1]-=- a quadratic polynomial has at most one non-repelling cycle in C. In particular, there is at most one cycle of Siegel disks. Proposition 2.7. Let c∗ ∈Mbe a parameter value for which fc has a Siegel di... |

80 |
Iterations of analytic functions
- Siegel
- 1942
(Show Context)
Citation Context ...e). An extreme example of a number of bounded type is the golden mean √ 5 − 1 θ∗ = =[1, 1, 1,...]. 2 The set D(2+) ≡ � k>2 has full measure in the interval [0, 1). In 1942 Siegel showed: Theorem 2.9 (=-=[Sie]-=-). Let R be an analytic map with a periodic point z0 ∈ Ĉ of period p. Suppose the multiplier of the cycle is λ = e 2πiθ with θ ∈D(2+). Then the local linearization equation (2.2) holds. Dks558 M. BRAV... |

59 |
Extension de la Notion de Fonction Récursive aux Fonctions d’une ou Plusieurs Variables Réelles III, Comptes Rendus de l’Académie des sciences Paris 241
- Lacombe
- 1955
(Show Context)
Citation Context ...n oracle TM M φ (m) such that if φ is an oracle for x ∈ [a,b], then on input m, M φ outputs a y ∈ D such that |y − f(x)| < 2 −m . This definition was first introduced by Grzegorczyk [Grz] and Lacombe =-=[Lac]-=- and follows in the tradition of Computable Analysis originated by Banach and Mazur in 1936–1939 (see [Maz]). To understand the definition better, the reader without a Computer Science background shou... |

46 |
Analytic form of differential equations
- Brjuno
- 1972
(Show Context)
Citation Context ...iθ with θ ∈D(2+). Then the local linearization equation (2.2) holds. Dks558 M. BRAVERMAN AND M. YAMPOLSKY The strongest known generalization of this result was proved by Brjuno in 1972: Theorem 2.10 (=-=[Bru]-=-). Suppose (2.4) B(θ)= � log(qn+1) < ∞. Then the conclusion of Siegel’s Theorem holds. n Note that a quadratic polynomial with a fixed Siegel disk with rotation angle θ after an affine change of coord... |

35 |
Does a Julia Set Depend Continuously on the Polynomial
- Douady
- 1994
(Show Context)
Citation Context ...H between compact sets X, Y in the plane (2.1). It turns out that the dependence c ↦→ J(fc) is discontinuous in the Hausdorff distance. For an excellent survey of this problem see the paper of Douady =-=[Do]-=-. The discontinuity which has found the most interesting dynamical applications occurs at parameter values for which fc has a parabolic point. We, however, will employ a more obvious discontinuity whi... |

24 |
Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque 231
- Yoccoz
- 1995
(Show Context)
Citation Context ...l’s Theorem holds. n Note that a quadratic polynomial with a fixed Siegel disk with rotation angle θ after an affine change of coordinates can be written as (2.5) Pθ(z)=z 2 + e 2πiθ z. In 1987 Yoccoz =-=[Yoc]-=- proved the following converse to Brjuno’s Theorem: Theorem 2.11 ([Yoc]). Suppose that for θ ∈ [0, 1) the polynomial Pθ has a Siegel point at the origin. Then B(θ) < ∞. The numbers satisfying (2.4) ar... |

23 |
The computational complexity of some Julia sets
- Rettinger, Weihrauch
- 2003
(Show Context)
Citation Context .... Namely C is the set of finite unions of dyadic balls: � n� � C = B(di,ri) | where di,ri ∈ D . i=1 The following definition is equivalent to the set computability definition given in [Wei] (see also =-=[RW]-=-). Definition 2.3. WesaythatacompactsetK ⊂ R k is computable if there exists a TM M(m), such that on an input m ∈ N,themachineM(m) outputs an encoding of Cm ∈Csuch that dH(K,Cm) < 2 −m . To illustrate... |

20 | The Brjuno functions and their regularity properties
- Marmi, Moussa, et al.
- 1997
(Show Context)
Citation Context ...mproved this result by showing that: Theorem 2.12 ([BC2]). The function θ ↦→ Φ(θ)+logr(θ) extends to R as a 1periodic continuous function. We remark that the following stronger conjecture exists (see =-=[MMY]-=-): Marmi-Moussa-Yoccoz Conjecture. [MMY] The function θ ↦→ Φ(θ) + log r(θ) is Hölder of exponent 1/2. qn θnsNON-COMPUTABLE JULIA SETS 559 Dependence of the conformal radius of a Siegel disk on the par... |

15 |
On the Julia set of a typical quadratic polynomial with a Siegel disk
- PETERSEN, ZAKERI
(Show Context)
Citation Context ...xamples. The size of the set of parameter values θ ∈ R/Z for which J(Pθ) isnoncomputable is rather meagre. One can show combining the results of [BBY1] with, for example, those of Petersen and Zakeri =-=[PZ]-=- that this set has Lebesgue measure zero; and Theorem 5.1 implies that its complement contains a dense Gδ subset of R/Z. It is natural to ask if, for example, its Hausdorff dimension is positive, and ... |

14 | Filled Julia sets with empty interior are computable
- Binder, Braverman, et al.
(Show Context)
Citation Context ...ther the construction of non-computable Julia sets carried out in this paper can be replaced with a different, perhaps, simpler approach. Jointly with I. Binder, we have demonstrated the following in =-=[BBY1]-=-: Theorem 5.1 ([BBY1]). Let R be a rational mapping of the Riemann sphere with no rotation domains (either Siegel disks or Herman rings). Then its Julia set is computable by a TM M φ with an oracle fo... |

14 |
Computable functionals, Fund
- Grzegorczyk
- 1955
(Show Context)
Citation Context ... if there exists an oracle TM M φ (m) such that if φ is an oracle for x ∈ [a,b], then on input m, M φ outputs a y ∈ D such that |y − f(x)| < 2 −m . This definition was first introduced by Grzegorczyk =-=[Grz]-=- and Lacombe [Lac] and follows in the tradition of Computable Analysis originated by Banach and Mazur in 1936–1939 (see [Maz]). To understand the definition better, the reader without a Computer Scien... |

12 | Computational Complexity of Euclidean Sets: Hyperbolic Julia Sets are Poly-Time Computable
- Braverman
- 2005
(Show Context)
Citation Context ... 2−m . InthecaseofJuliasets: Definition 2.5. We say that Jc is computable if the function J : d ↦→ Jd is computable on the set {c}.s554 M. BRAVERMAN AND M. YAMPOLSKY The following has been shown (see =-=[Brv1]-=-, [Ret]): Theorem 2.3. Denote by H the set of parameters c for which Jc is hyperbolic. Then (i) Jc is computable for all c ∈H; moreover, (ii) the function J is computable on each bounded subset of H. ... |

12 | Parabolic Julia sets are polynomial time computable
- Braverman
(Show Context)
Citation Context ...rbitrarily high, but again all with Siegel disks. A natural first step towards studying the complexity of Cremer Julia sets is to look at parabolics, but the first author has recently demonstrated in =-=[Brv2]-=- that having a parabolic orbit does not qualitatively change the complexity of computing a Julia set. This opens an entertaining possibility that some Cremer Julia sets have attainable computational c... |

12 |
Petits diviseurs en dimension 1
- Yoccoz
- 1995
(Show Context)
Citation Context ...ABLE JULIA SETS 9 n Note that a quadratic polynomial with a fixed Sigel disk with rotation angle θ after an affine change of coordinates can be written as (2.5) Pθ(z) = z 2 + e 2πiθ z. In 1987 Yoccoz =-=[Yoc]-=- proved the following converse to Brjuno’s Theorem: Theorem 2.11 ([Yoc]). Suppose that for θ ∈ [0, 1) the polynomial Pθ has a Siegel point at the origin. Then B(θ) < ∞. The numbers satisfying (2.4) ar... |

10 | On computational complexity of Siegel Julia sets
- Binder, Braverman, et al.
(Show Context)
Citation Context ...the computational complexity of these sets (the amount of time it takes to decide whether to color a pixel of size 2−n as a function of n) is very high. This is indeed so for the naïve algorithms. In =-=[BBY2]-=- jointly with I. Binder we have constructed quadratic Julia sets whose computational complexity is arbitrarily high, but again all with Siegel disks. A natural first step towards studying the complexi... |

9 |
Classes récursivement fermés et fonctions majorantes. Comptes Rendus Académie des Sciences
- Lacombe
- 1955
(Show Context)
Citation Context ... oracle TM M φ (m) such that if φ is an oracle for x ∈ [a, b], then on input m, M φ outputs a y ∈ D such that |y − f(x)| < 2 −m . This definition was first introduced by Grzegorczyk [Grz] and Lacombe =-=[Lac]-=-, and follows in the tradition of Computable Analysis originated by Banach and Mazur in 1937 (see [Maz]).NON-COMPUTABLE JULIA SETS 3 To understand the definition better, the reader without a Computer... |

8 |
Polynomial-time computability in analysis, in “Handbook of Recursive Mathematics”, Volume 2
- Ko
- 1998
(Show Context)
Citation Context ...onstant c; hence υ has an (easily) computable modulus of continuity. These two facts together imply that υ is computable by a single machine of the interval [0, 1] (see for example Proposition 2.6 in =-=[Ko2]-=-). This implies the Conditional Implication. The following conditional result follows: Lemma 2.21 (Conditional). Suppose the Conditional Implication holds. Let θ ∈ [0, 1] be such that Φ(θ) is finite. ... |

6 |
Quadratic Siegel disks with smooth boudaries
- Buff, Chéritat
(Show Context)
Citation Context ...nction Φ(θ) is continuous on each Ωi, i ∈ N, and hence so is r(θ)=exp(υ(θ) − Φ(θ)). By our assumption, ∞� R/Z = Ωi, and we arrive at a contradiction with Theorem 2.22. � A note on the connection with =-=[BC1]-=-. A. Chéritat has pointed out to us that the methods of [BC1], where Siegel disks with smooth boundaries are constructed for the quadratic family, can be used to derive the Main Theorem. We discuss th... |

5 |
Siegel disks with smooth boundaries
- Avila, Buff, et al.
(Show Context)
Citation Context ...the work of Buff and Chéritat. Let us outline here how the methods of [BC1] can be applied to prove Theorem 2.22 instead of the estimates of §3 (we note that a newer version of the same result exists =-=[ABC]-=-, where the arguments we quote are simplified). The main technical result of that paper is the following. Let α =[a0,a1,...] be a Brjuno number, and as before denote by pk/qk the sequence of its conti... |

5 |
Variation of the conformal radius
- Rohde, Zinsmeister
(Show Context)
Citation Context ...he proposition follows from this and the Carathéodory Kernel Theorem. � In fact, we can state the following quantitative version of the above result. For the proof, based on Koebe’s Theorem, see e.g. =-=[RZ]-=-. Lemma 2.15. Let U be a simply-connected bounded subdomain of C containing the point 0 in the interior. Suppose V ⊂ U is a simply-connected subdomain of U, and ∂V ⊂ Uɛ(∂U). Then 0 <r(U, 0) − r(V,0) ≤... |

3 |
The Yoccoz function continuously estimates the size of Siegel disks
- Buff
(Show Context)
Citation Context ...east r(θ)/4. Yoccoz [Yoc] has shown that the sum Φ(θ)+logr(θ) is bounded from below independently of θ ∈B. Recently, Buff and Chéritat have greatly improved this result by showing that: Theorem 2.12 (=-=[BC2]-=-). The function θ ↦→ Φ(θ)+logr(θ) extends to R as a 1periodic continuous function. We remark that the following stronger conjecture exists (see [MMY]): Marmi-Moussa-Yoccoz Conjecture. [MMY] The functi... |

3 |
Computable analysis, Rosprawy Matematyczne
- Mazur
- 1963
(Show Context)
Citation Context ...at |y − f(x)| < 2 −m . This definition was first introduced by Grzegorczyk [Grz] and Lacombe [Lac] and follows in the tradition of Computable Analysis originated by Banach and Mazur in 1936–1939 (see =-=[Maz]-=-). To understand the definition better, the reader without a Computer Science background should think of a computer program with an instruction READ real number x WITH PRECISION n(m). On the execution... |

2 |
A fast algorithm for Julia sets of hyperbolic rational functions
- Retinger
- 2005
(Show Context)
Citation Context ...nthecaseofJuliasets: Definition 2.5. We say that Jc is computable if the function J : d ↦→ Jd is computable on the set {c}.s554 M. BRAVERMAN AND M. YAMPOLSKY The following has been shown (see [Brv1], =-=[Ret]-=-): Theorem 2.3. Denote by H the set of parameters c for which Jc is hyperbolic. Then (i) Jc is computable for all c ∈H; moreover, (ii) the function J is computable on each bounded subset of H. Our goa... |