## Is the Mandelbrot set computable? (2005)

Venue: | MATH. LOGIC QUART |

Citations: | 10 - 0 self |

### BibTeX

@ARTICLE{Hertling05isthe,

author = {Peter Hertling},

title = {Is the Mandelbrot set computable?},

journal = {MATH. LOGIC QUART},

year = {2005},

volume = {51},

pages = {5--18}

}

### OpenURL

### Abstract

We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two–sided distance function of the Mandelbrot set is computable if the hyperbolicity conjecture is true. We formulate the question whether the distance function of the Mandelbrot set is computable also in terms of the escape time.

### Citations

838 |
Theory of Recursive Functions and Effective Computability
- ROGERS
- 1967
(Show Context)
Citation Context ... This notion is used in the usual sense. The computability of functions f :⊆sn →scan be defined via Turing machines or via the calculus of µ-recursive functions or in many other ways; see e.g. Rogers =-=[26]-=-, Soare [29], Weihrauch [31]. We shall use the computable bijection 〈., .〉s: 2s→ defined by 〈i, j〉 := 1 (i + j) · (i + j + 1) + j. One defines 2 bijective pairing functions 〈., . . . , .〉s: ns→ for n ... |

476 |
Recursively enumerable sets and degrees
- Soare
- 1987
(Show Context)
Citation Context ... is used in the usual sense. The computability of functions f :⊆sn →scan be defined via Turing machines or via the calculus of µ-recursive functions or in many other ways; see e.g. Rogers [26], Soare =-=[29]-=-, Weihrauch [31]. We shall use the computable bijection 〈., .〉s: 2s→ defined by 〈i, j〉 := 1 (i + j) · (i + j + 1) + j. One defines 2 bijective pairing functions 〈., . . . , .〉s: ns→ for n ≥ 3 by induc... |

377 |
On a theory of computation and complexity over the real numbers
- Blum, Shub, et al.
- 1989
(Show Context)
Citation Context ...e first computability notion over the real numbers which we wish to discuss is the more algebraically oriented computability notion described by the real number machine model by Blum, Shub, and Smale =-=[3, 2]-=-. In this model one computes in an algebraic way with real numbers, assuming that they are given with infinite precision, and that one can perform the usual algebraic operations +, −, ∗, /, and compar... |

348 |
Complexity and Real Computation
- Blum, Cucker, et al.
- 1998
(Show Context)
Citation Context ...e first computability notion over the real numbers which we wish to discuss is the more algebraically oriented computability notion described by the real number machine model by Blum, Shub, and Smale =-=[3, 2]-=-. In this model one computes in an algebraic way with real numbers, assuming that they are given with infinite precision, and that one can perform the usual algebraic operations +, −, ∗, /, and compar... |

237 |
Computable Analysis
- Weihrauch
- 2000
(Show Context)
Citation Context ...hich take this numerical point of view into account, and according to which the epigraph of the exponential function is computable. These notions are studied in computable analysis; compare Weihrauch =-=[33]-=-. In Section 2 we shall introduce and discuss them in detail. The basic idea is that one tries to obtain a good rational approximation of the result by using rational approximations of the (ideal) rea... |

163 |
Complexity theory of real functions
- Ko
- 1991
(Show Context)
Citation Context ...eal functions considered in computable analysis. It is based on work by Grzegorczyk [13] and Lacombe [19]. Recent monographs concerned with this computability notion are Pour–El and Richards [24], Ko =-=[17]-=-, and Weihrauch [33]. A function f :⊆ ¡ n → ¡ is Lipschitz-continuous if there exists a positive constant c such that d(f(x), f(y)) ≤ c · d(x, y) for all x, y ∈ dom(f). For Lipschitz–continuous functi... |

155 |
Complex dynamics
- Carleson, Gamelin
- 1993
(Show Context)
Citation Context ...Propositions 9, 11, and 8. Since Φ is holomorphic, also the derivative of Φ is computable (Pour-El and Richards [24]). For explicit formulas for Φ see Jungreis [16], Branner [5], Carleson and Gamelin =-=[8]-=-. Computations based on Φ and Φ ′ resp. on the potential function G(c) := log(|Φ(c)|) of M (where log denotes the natural logarithm) and its derivative G ′ have been very useful for actual high precis... |

139 |
The Emperor’s New Mind. Concerning computers, minds, and the laws of physics
- Penrose
- 1990
(Show Context)
Citation Context ...iate quadratic complex polynomials in the complex plane. Date: March 26, 2003. 1s2 PETER HERTLING In this paper we are concerned with the following question: Is the Mandelbrot set computable? Penrose =-=[23]-=- raised this question. Of course, before one can try to answer this question, one has to make up one’s mind about the meaning of the word “computable”. We wish to consider a mathematically precise com... |

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 ... of f to some subset of dom(f) is computable. This notion is one of the computability notions for real functions considered in computable analysis. It is based on work by Grzegorczyk [13] and Lacombe =-=[19]-=-. Recent monographs concerned with this computability notion are Pour–El and Richards [24], Ko [17], and Weihrauch [33]. A function f :⊆ ¡ n → ¡ is Lipschitz-continuous if there exists a positive cons... |

58 |
Computable functions
- Grzegorczyk
- 1955
(Show Context)
Citation Context ...o any restriction of f to some subset of dom(f) is computable. This notion is one of the computability notions for real functions considered in computable analysis. It is based on work by Grzegorczyk =-=[13]-=- and Lacombe [19]. Recent monographs concerned with this computability notion are Pour–El and Richards [24], Ko [17], and Weihrauch [33]. A function f :⊆ ¡ n → ¡ is Lipschitz-continuous if there exist... |

54 |
Itération des polynômes quadratiques complexes
- Douady, Hubbard
- 1982
(Show Context)
Citation Context ...andelbrot set. set contained in the closed disk of radius 2 around the origin. Furthermore, it is equal to the closure of its interior, and it is connected and simply connected, as Douady and Hubbard =-=[9]-=- have shown. For an overview of these and other properties of the Mandelbrot set the reader is referred to Branner [5]. The importance of the Mandelbrot set stems from the fact that it describes the d... |

49 |
Computability on subsets of Euclidean space I: Closed and compact subsets
- Brattka, Weihrauch
- 1999
(Show Context)
Citation Context ... Real Numbers In this section we introduce basic notions from Computable Analysis, following to a large extent Weihrauch [33]. For the topics treated here one might also consult Brattka and Weihrauch =-=[7]-=- or Hertling [15]. After introducing some notation we will introduce computability notions over the real numbers which correspond to the classical computability notions over the natural numbers: compu... |

47 | Applied and Computational Complex Analysis. Volume 2 - Henrici - 1974 |

45 | Periodic orbits, externals rays and the Mandelbrot set: an expository account (Géométrie complexe et systèmes dynamiques
- Milnor
- 1995
(Show Context)
Citation Context ... i is given. One can compute the center ci and the period, say, k, of the attracting cycle of pci. Let H(M) (k) be the union of all (finitely many; one can easily compute the exact number; see Milnor =-=[21]-=- or Schleicher [27]) hyperbolic components such that this cycle has period k. By a similar reasoning as in the proof of Proposition 10 one can see that one can enumerate a list L1 of rational balls co... |

44 | The Hausdorff dimension of the boundary of the Mandelbrot set and Julia
- Shishikura
- 1998
(Show Context)
Citation Context ...sidered a subset of ¡ 2 , as was shown by Blum and Smale [4]. They showed that, if the Mandelbrot set were BSS–decidable, its boundary would have to have Hausdorff dimension at most 1. But Shishikura =-=[28]-=- has shown that the boundary of M has Hausdorff dimension 2. Hence, M is not BSS–decidable. Thus, one may regard the question whether the Mandelbrot set is computable as settled. But, also the epigrap... |

38 |
Constructive analysis, volume 279 of Grundlehren der Mathematischen Wissenschaften
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ... is computable; see e.g. Brattka [6]. The idea to consider sets A with effective distance function dA goes back to constructive analysis. There, such sets are called “located”; see Bishop and Bridges =-=[1]-=-. Ge and Nerode [12] used this notion in the framework of computable analysis and called it “Turing locatedness”. It is a fundamental fact that a set A of natural numbers is decidable if, and only if,... |

23 |
The computational complexity of some Julia sets
- Rettinger, Weihrauch
- 2003
(Show Context)
Citation Context ... Julia sets are computable in the sense of computable analysis. Once a set has turned out to be computable, the next natural question is: what is its computational complexity? Rettinger and Weihrauch =-=[25]-=- have obtained first results concerning the computational complexity of certain computable Julia sets. 2. Computability over the Real Numbers In this section we introduce basic notions from Computable... |

21 |
Computability, volume 9
- Weihrauch
- 1987
(Show Context)
Citation Context ...usual sense. The computability of functions f :⊆sn →scan be defined via Turing machines or via the calculus of µ-recursive functions or in many other ways; see e.g. Rogers [26], Soare [29], Weihrauch =-=[31]-=-. We shall use the computable bijection 〈., .〉s: 2s→ defined by 〈i, j〉 := 1 (i + j) · (i + j + 1) + j. One defines 2 bijective pairing functions 〈., . . . , .〉s: ns→ for n ≥ 3 by induction: 〈i1, . . .... |

20 |
The Mandelbrot Set
- Branner
- 1989
(Show Context)
Citation Context ... of its interior, and it is connected and simply connected, as Douady and Hubbard [9] have shown. For an overview of these and other properties of the Mandelbrot set the reader is referred to Branner =-=[5]-=-. The importance of the Mandelbrot set stems from the fact that it describes the dynamic behaviour of all univariate quadratic complex polynomials in the complex plane. Date: March 26, 2003. 1s2 PETER... |

16 |
The Godel incompleteness theorem and decidability over a ring
- Blum, Smale
- 1990
(Show Context)
Citation Context ...of Blum, Shub, and Smale which computes the characteristic function of A. In this model the Mandelbrot set turns out to be noncomputable if considered a subset of ¡ 2 , as was shown by Blum and Smale =-=[4]-=-. They showed that, if the Mandelbrot set were BSS–decidable, its boundary would have to have Hausdorff dimension at most 1. But Shishikura [28] has shown that the boundary of M has Hausdorff dimensio... |

16 |
Self-similarity and hairiness in the Mandelbrot set
- Milnor
- 1989
(Show Context)
Citation Context ... of a point outside 4 M from M in terms of Φ and its derivative Φ ′ resp. in terms of G and G ′ : (3) sinh G(c) 2 · eG(c) · |G ′ (c)| < dM(c) 2 · sinh G(c) < |G ′ (c)| for all c ∈s\ M; compare Milnor =-=[20]-=-, Fisher [11]. Using a recursion scheme one can compute G(c) and G ′ (c) even quite fast. This is due to the fact that G(c) = limn→∞(log(|p◦n+1 c (0)|)/2n ) for all c ∈sFinally, the function \ M. �e(c... |

16 |
Computability on computable metric spaces
- Weihrauch
- 1993
(Show Context)
Citation Context ...ed informally after the lemma are realizations of this idea. The first two conditions in the lemma are simply effective versions of continuity.sIS THE MANDELBROT SET COMPUTABLE? 7 Lemma 4. (Weihrauch =-=[32]-=-) For a function f :⊆ ¡ n → ¡ m the following three conditions are equivalent. 1. There is a c.e. set A ⊆s2 such that f(dom(f) ∩ closure(B n (i))) ⊆ B m (j) for all (i, j) ∈ A, and for every x ∈ dom(f... |

12 |
Nerode: “On Extreme Points of Convex Compact Turing Located Sets
- Ge, A
- 1994
(Show Context)
Citation Context ... e.g. Brattka [6]. The idea to consider sets A with effective distance function dA goes back to constructive analysis. There, such sets are called “located”; see Bishop and Bridges [1]. Ge and Nerode =-=[12]-=- used this notion in the framework of computable analysis and called it “Turing locatedness”. It is a fundamental fact that a set A of natural numbers is decidable if, and only if, it is c.e. and its ... |

10 |
Recursively enumerable subsets of R q in two computing models: BlumShub -Smale machine and Turing machine
- Zhong
(Show Context)
Citation Context ...nalysis, which are introduced in the following section, lead to natural and interesting questions also in the context of complex dynamical systems. In this context we would like to mention that Zhong =-=[34]-=- has compared the computability notions for subsets of Euclidean space from the Blum–Shub–Smale theory and from computable analysis and has proved that certain Julia sets are computable in the sense o... |

9 |
Classes récursivement fermés et fonctions majorantes. Comptes Rendus Académie des Sciences
- Lacombe
- 1955
(Show Context)
Citation Context ...there is an algorithm which, given a ρ n -name of a point x ∈ ¡ n , halts after finitely many computation steps if, and only if, x lies in U. The notion of c.e. openness goes back at least to Lacombe =-=[18]-=-. For example the empty set and the whole space ¡ n are c.e. open and c.e. closed. The following lemma gives a simple connection between the two c.e. conditions. Lemma 2. The closure of a c.e. open se... |

8 | An effective Riemann Mapping Theorem
- Hertling
- 1999
(Show Context)
Citation Context ... this section we introduce basic notions from Computable Analysis, following to a large extent Weihrauch [33]. For the topics treated here one might also consult Brattka and Weihrauch [7] or Hertling =-=[15]-=-. After introducing some notation we will introduce computability notions over the real numbers which correspond to the classical computability notions over the natural numbers: computably enumerable ... |

7 | The emperor’s new recursiveness: the epigraph of the exponential function in two models of computability - Brattka - 2000 |

6 | Etude dynamique des polynômes complexes (Deuxième partie - Douady, Hubbard - 1985 |

5 | Rational parameter rays of the Mandelbrot set
- Schleicher
(Show Context)
Citation Context ...n compute the center ci and the period, say, k, of the attracting cycle of pci. Let H(M) (k) be the union of all (finitely many; one can easily compute the exact number; see Milnor [21] or Schleicher =-=[27]-=-) hyperbolic components such that this cycle has period k. By a similar reasoning as in the proof of Proposition 10 one can see that one can enumerate a list L1 of rational balls covering exactly H(M)... |

3 |
Exploring the mandelbrot set
- Fisher
- 1988
(Show Context)
Citation Context ...utside 4 M from M in terms of Φ and its derivative Φ ′ resp. in terms of G and G ′ : (3) sinh G(c) 2 · eG(c) · |G ′ (c)| < dM(c) 2 · sinh G(c) < |G ′ (c)| for all c ∈s\ M; compare Milnor [20], Fisher =-=[11]-=-. Using a recursion scheme one can compute G(c) and G ′ (c) even quite fast. This is due to the fact that G(c) = limn→∞(log(|p◦n+1 c (0)|)/2n ) for all c ∈sFinally, the function \ M. �e(c) := − log 2 ... |

3 |
The fundamental theorem of algebra in recursive analysis
- Specker
- 1969
(Show Context)
Citation Context ...e 〈k, m〉, for each fixed point z0 of p ◦k c , it computes a rational 2 −m –approximation qz0,m of (p ◦k c ) ′ (z0) (note that using an effective version of the Fundamental Theorem of Algebra (Specker =-=[30]-=-) one can compute the fixed points of p ◦k c , that is, the zeros of p ◦k c (z) − z, with any required precision), and checks whether the absolute value of this rational complex number is smaller than... |

2 |
The uniformization of the complement of the Mandelbrot set
- JUNGREIS
- 1985
(Show Context)
Citation Context ...s inverse are computable. Proof. Due to Propositions 9, 11, and 8. Since Φ is holomorphic, also the derivative of Φ is computable (Pour-El and Richards [24]). For explicit formulas for Φ see Jungreis =-=[16]-=-, Branner [5], Carleson and Gamelin [8]. Computations based on Φ and Φ ′ resp. on the potential function G(c) := log(|Φ(c)|) of M (where log denotes the natural logarithm) and its derivative G ′ have ... |