IS THE MANDELBROT SET COMPUTABLE?
PETER HERTLING
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.
1. Introduction
Let us consider the following class of univariate polynomials
pc (z) := z 2 + c,
for arbitrary complex numbers c ∈ . We denote the k–th iterate of pc , for k ≥ 0,
◦0
by p◦k
. The set
c , where pc is the identity function on
M := {c ∈
| p◦k
c (0) 6→ ∞ for k → ∞}
is called the Mandelbrot set; see Figure 1. It is easy to see that M is a compact
Figure 1. The Mandelbrot 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 dynamic behaviour of all
univariate quadratic complex polynomials in the complex plane.
Date: March 26, 2003.
1
2
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 computability notion.
Which subsets of the complex plane should be called “computable”? Currently,
several nonequivalent computability notions over the real or complex numbers are
being considered. This is in contrast to the discrete case. There is a standard,
universally accepted computability notion for functions and sets over the natural
numbers. It is part of computability theory, also called recursion theory, and can be
defined via the Turing machine model. For example, a set A ⊆ is called decidable,
if the characteristic function of A is computable, that is, if there is a Turing machine
(to be precise: given the binary
which, given an arbitrary natural number n ∈
name of n) stops after finitely many steps and tells whether n is an element of A or
not.
The 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 comparisons
on real numbers with infinite precision. Let us call a subset A ⊆ n for some n ≥ 1
BSS–decidable if there is a real number machine in the sense 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 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 epigraph {(x, y) ∈ 2 | exp(x) ≤ y} of the
exponential function is not BSS–decidable (Brattka [6]). This is in contrast to the
observation that in a numerical sense it is quite a simple subset of the real plane.
Indeed, it is very easy to draw a picture of it with any desired precision. Compare
also Brattka [6] and Weihrauch [33, Chapter 9] on this topic.
In the rest of the paper we shall consider computability notions which 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) real input. The formal definitions are
based on the one hand on computability theory and on the other hand on topology.
We shall see that these computability notions for functions and for sets over the real
numbers are natural generalizations of the fundamental computability notions over
the natural numbers: of computable functions, computably enumerable sets (also
called recursively enumerable sets), and decidable sets (also called computable sets).
It has turned out that there are at least two different effectivity notions for sets of real
numbers which correspond to computable enumerability of sets of natural numbers,
IS THE MANDELBROT SET COMPUTABLE?
3
one for open subsets of n and one for closed subsets of n. We shall call the corresponding sets c.e. open resp. c.e. closed. As for decidability, it has turned out that
the perhaps first idea to demand again computability of the characteristic function,
does not lead to a useful computability notion. This is due to the approximative
nature of our computability notion for functions over the real numbers. But, if one
demands computability not of the characteristic function but, instead, of the distance
function or the two-sided distance function of a closed set then one arrives at natural
generalizations of the notion of a decidable set of natural numbers. In Section 2 we
also prove that there is a proper hierarchy of sets which is interesting in connection
with sets like the Mandelbrot set, and we formulate a corollary of an effective version
of the Riemann mapping theorem which we will need later on.
In Section 3 we state and prove the results of this paper concerning computability
of the Mandelbrot set and of related sets. We give a proof of the well-known fact that
the exterior of the Mandelbrot set is c.e. open. We show that a certain, prominent
subset of its interior, the union of the hyperbolic components, is c.e. open, and that
the boundary of the Mandelbrot set is c.e. closed. We conclude that the two-sided
distance function of the Mandelbrot set is computable if the hyperbolicity conjecture
is true. In that case, also the (usual) distance function of the Mandelbrot set is
computable. We formulate the question whether the (usual) distance function of the
Mandelbrot set is computable in terms of the escape time. In connection with this
question, it is interesting that the natural conformal function Φ, defined by Douady
and Hubbard [9], which maps the exterior of the Mandelbrot set onto the exterior
of the unit disk, is computable. Finally, we show that the hyperbolic components of
the interior of the Mandelbrot set are uniformly computable in a strong sense.
The proofs are not deep and rely only on well-known facts about the Mandelbrot
set, most of them contained in the introductory text [5] by Branner. We hope to show
that the recursion-theoretic computability notions of computable analysis, 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 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 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
sets, computable functions, and computable sets. These are the notions which we
need in order to formulate our results and conjectures about the computability of
the Mandelbrot set. Then we prove that there is a proper hierarchy of sets which
4
PETER HERTLING
is interesting in connection with sets like the Mandelbrot set. Finally, we formulate
a corollary of an effective version of the Riemann mapping theorem which will be
useful later on.
2.1. Notation and Basic Notions. First, we introduce some notation. By we
denote the set of natural numbers including 0, that is, = {0, 1, 2, . . . }. By f :⊆
X → Y we denote a possibly partial function with domain dom(f ) contained in X
and with range contained in Y . If f is total, that is, if dom(f ) = X, we may indicate
this by writing f : X → Y . A sequence of elements of a set X is a total function
x : → X, and will be written (xi )i .
All of the computability notions introduced and used in this paper are based on
the notion of a computable function f :⊆ n →
for n ≥ 1. This notion is used
in the usual sense. The computability of functions f :⊆ n →
can 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 h., .i : 2 → defined by hi, ji := 21 (i + j) · (i + j + 1) + j. One defines
bijective pairing functions h., . . . , .i : n → for n ≥ 3 by induction: hi1 , . . . , in i :=
hhi1 , . . . , in−1 i, in i. We also define h·i : → by hii := i. Note that for n ≥ 1 a
function f :⊆ n → is computable if, and only if, the function g :⊆ → defined
by g(hi1 , . . . , in i) := f (i1 , . . . , in ) is computable.
We wish to perform computations on real numbers in an approximative sense.
Therefore, we will approximate real numbers by rational numbers and perform the
actual computations on rational numbers. We can translate computability from natural numbers to rational numbers by using for example the numbering ν : →
i−j
of rational numbers defined by ν hi, j, ki := k+1
, and the numbering ν n : → n of
rational vectors defined by ν n hl1, . . . , ln i := (ν (l1), . . . , ν (ln )i. By d : n → we
denote the usual Euclidean distance on n. Sometimes we use |x| for the Euclidean
length (norm) of x ∈ n. The open ball in n with center x and radius r is written
B(x, r) := {y ∈ n | d(x, y) < r}. Then, radius(B(x, r)) = r. It will also be convenient to fix a numbering Bn of all nonempty rational open balls (open balls with
j+1
).
rational center and positive rational radius) in n: Bn (hi, hj, kii) := B(ν n (i), k+1
A point y will sometimes be called an ε-approximation of a point x if d(x, y) ≤ ε.
Let us fix some n ≥ 1. We call a sequence (mi)i of natural numbers a ρn -name of
a point x ∈ n if ν n (mi ) is a 2−i -approximation of x, for each i. A point x ∈ n
is called computable if it has a computable ρn –name. Similarly, a sequence (xi )i of
points xi ∈ n is called computable if there is a computable function f : 2 →
such that ν n (f (i, k)) is a 2−k –approximation of xi , for all i, k.
Let := {p | p : → } be the set of all sequences of natural numbers. Although
we do not need it for the formulation of the main results, it is convenient to have a
definition of computability also for functions F :⊆ →
and for functions G :⊆
→ . The idea behind the computability notion for functions F :⊆ → is that
given a sufficiently long prefix of p ∈ dom(F ) one should be able to compute the value
F (p). In other words, a function F :⊆ → is computable if there is an algorithm
(a Turing machine) which, given the elements of a sequence p ∈ dom(F ) one by one,
is able to compute F (p) after finitely many steps. For functions G :⊆ → the
IS THE MANDELBROT SET COMPUTABLE?
5
idea is similar: one should be able to compute any component G(p)i of G(p) when
one knows a sufficiently long prefix of p ∈ dom(G). The length of the prefix may and
usually does depend on p and on i. These notions are treated in detail e.g. by Rogers
[26] and Weihrauch [31]. For completeness sake we give a formal definition. We use
the following bijection b : ∗ → between the set ∗ of all finite strings over and
:
b(empty string) := 0,
b(i0 , . . . , ik ) := 1 + hk, hi0 , . . . , ik ii.
A function F :⊆ → is computable if there is a computable function f :⊆ →
with the following properties:
1. if b(i0 , . . . , ik−1 ) ∈ dom(f ), and for some l > k and ik , . . . , il−1 ∈
also
b(i0 , . . . , il−1 ) ∈ dom(f ), then f b(i0 , . . . , ik−1 ) = f b(i0 , . . . , il−1).
2. dom(F ) = {p ∈ | (∃k ∈ ) b(p0 , . . . , pk−1 ) ∈ dom(f )}.
3. If p ∈ dom(F ) and k is a number such that b(p0 , . . . , pk−1 ) ∈ dom(f ) then
F (p) = f (b(p0 , . . . , pk−1 )).
→
is computable if there is a computable function F :⊆
A function G :⊆
→
such that dom(G) = {p ∈
| (∀n ∈ ) (n, p) ∈ dom(F )} and G(p) =
(F (0, p), F (1, p), F (2, p), . . . ) for all p ∈ dom(G). Here, by (n, p) we mean the sequence (n, p0 , p1 , p2 , . . . ).
2.2. Computably Enumerable Sets. Besides the notion of a computable function
f :⊆ n → there are two other computability notions over the natural numbers
which are of fundamental importance: the notion of a computably enumerable set
A ⊆ n and the notion of a decidable or computable set A ⊆ . We wish to define
analogous notions over the real numbers. In this subsection we consider computably
enumerable sets. A set A ⊆ is called computably enumerable (c.e.) if, and only if,
one and then all of the following three equivalent conditions are fulfilled:
1. there is a computable function f :⊆ → with dom(f ) = A,
2. there is a computable function f :⊆ → whose range is equal to A,
3. either A is empty or there is a total computable function f :
→
whose
range is equal to A.
Using the pairing function h , i one can translate this notion to subsets of n .
When one tries to define an analogous notion for subsets of n one realizes that
there topology plays an important role and that there are (at least) two different
natural generalizations of computable enumerability for subsets of n, one for open
sets and one for closed sets. For a proof of the following lemma see [7].
Lemma 1. 1. For an open set U ⊆ n the following three conditions are equivalent:
S
(a) There is a c.e. set A ⊆ with U = i∈A Bn (i).
(b) The set {i | closure(Bn (i)) ⊆ U } is computably enumerable.
(c) There is a computable function F :⊆
→
with the property that, for
n
n
a ρ -name p of a point x ∈
, the value F (p) is defined if, and only if,
x ∈ U.
2. For a closed set A ⊆ n the following two conditions are equivalent:
6
PETER HERTLING
(a) The set A is empty or there is a computable sequence (xn )n of points such
that the set {xn | n ∈ } is a dense subset of A.
(b) The set {i | Bn (i) ∩ A 6= ∅} is computably enumerable.
An open set U ⊆ n is called c.e. open if it satisfies one (and then all) of the first
three conditions. A closed set A ⊆ n is called c.e. closed if it satisfies one (and
then both) of the last two conditions. Stated informally, the third condition for c.e.
openness of U says that there is an algorithm which, given a ρn -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 set is c.e. closed.
Proof. The proof (see e.g. [15]) is easy: an open rational ball intersects the closure of
an open set U if, and only if, it contains a nonempty closed rational ball contained in
U . Thus, if one can enumerate all closed rational balls contained in U , then one can
also enumerate all open rational balls having nonempty intersection with the closure
of U .
It is easy to check that the union and the intersection of two c.e. open sets are
c.e. open again, and that the union of two c.e. closed sets is c.e. closed again. The
intersection of two c.e. closed sets does not need to be c.e. closed again, but at least
the following statement, which we will use later on, is true.
Lemma 3. If A ⊆ n and B ⊆ n are c.e. closed sets such that the boundary ∂B
of B is contained in A ∩ B, then A ∩ B is c.e. closed.
Proof. Let A ⊆ n and B ⊆ n be c.e. closed sets such that the boundary ∂B of B is
contained in A∩B. Then, an open rational ball Bn (i) has nonempty intersection with
A ∩ B if, and only if, it has nonempty intersection with A and nonempty intersection
with B. This follows because Bn (i) is path connected.
2.3. Computable Functions. In the following section we introduce computability
for subsets of n. For this, it is useful to have a computability notion for real number
functions. In fact, for our computability notion for subsets of n, it is sufficient to
define computability for Lipschitz continuous functions, and the reader who is not
interested in the general definition of computable real number functions is advised to
jump to the end of this subsection where we characterize computability for Lipschitz
continuous real number functions in a very simple way.
Which real number functions are computable? The computability notion which we
use is motivated by the demand that it should describe what is in principle possible
by numerical computation over the real numbers. The idea is that we call a function
f :⊆ n → m computable if one is able to compute the value f (x) of f at some point
x with some desired precision when one knows x with sufficient precision. This should
be possible for any desired output precision. The three conditions in the following
lemma and the condition formulated informally after the lemma are realizations of
this idea. The first two conditions in the lemma are simply effective versions of
continuity.
IS 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 ⊆ 2 such that f (dom(f ) ∩ closure(Bn (i))) ⊆ Bm (j) for
all (i, j) ∈ A, and for every x ∈ dom(f ) and every ε > 0 there is at least one
pair (i, j) ∈ A such that x ∈ Bn (i) and radius(Bm (j)) ≤ ε.
S
2. There is a c.e. set A ⊆ 2 such that f −1 (Bm (j)) = dom(f ) ∩ (i,j)∈A Bn (i) for
every j ∈ .
3. There is a computable function F :⊆ → such that for any ρn -name of any
point x ∈ dom(f ), the value F (p) exists and is a ρm -name of f (x).
A function f :⊆ n → m is called computable if it satisfies one and then all of
the conditions in Lemma 4. Another way to express this is as follows. A function
f :⊆ n → m is computable if, and only if, there is an algorithm (a Turing machine)
performing the following task for any point x in the domain of f : given a nonnegative
integer k, it may finitely often ask a question of the following kind “Give me a rational
point qj with d(x, qj ) ≤ 2−ij !” (where it has computed the number ij from k and
from the answers q0, . . . , qj−1 to the previous questions), and after finitely many
computation steps stops and produces a rational number r with d(f (x), r) ≤ 2−k .
Note that we do not demand that the domain of a computable function is determined by an algorithm for computing it. If some function f is computable then also
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 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
functions with c.e. open domain computability can be characterized in an especially
simple way. We say a function g :⊆ n × → is (ν n , id , ν )–computable if there
is a computable function h : × → such that g(ν n (i), k) = ν (h(n, k)) for all
i, k with (ν n (i), k) ∈ dom(g).
Lemma 5. Let U ⊆ n be c.e. open. Let f : U ⊆ n → be a Lipschitz–continuous
function with domain U . Then, f is computable if, and only if, there is a (ν n , id , ν )–
computable function g :⊆ n ×
→
with (U ∩ n ) ×
⊆ dom(g) satisfying
−k
n
d(f (q), g(q, k)) ≤ 2 for all q ∈
∩ U and k ∈ .
We omit the easy proof.
2.4. Computable Sets. A set A ⊆ is called decidable or computable if its characteristic function is computable, that means, if there is an algorithm which on input
n ∈ stops after finitely many steps and says YES or NO depending on whether n
is an element of A or not. We wish to obtain an analogous notion for sets A ⊆ n.
It is natural to try to introduce computability for subsets A ⊆ n in the same way
as decidability for subsets A ⊆ : by demanding that the characteristic function of
A is computable. But, using the computability notion for real functions introduced
above, this does not lead to a useful notion since for a boundary point x of A, a
8
PETER HERTLING
finite precision approximation to x will never suffice to decide whether x belongs to
A or not. According to this notion only the empty set and the full set n would be
computable. Indeed, computable functions over the real numbers, as defined above,
must be continuous. An idea which leads to a more useful computability notion for
subsets of n is to consider a “smooth” version of the characteristic function, for
example, the distance function dA : n → , defined by
dA (x) := inf d(x, y)
y∈A
n
for any nonempty subset A ⊆
and x ∈ n. We shall consider nonempty closed
subsets A ⊆ n as “computable” if the distance function dA is a computable function.
A slighly stronger computability notion is obtained if one considers the two–sided
n
distance function dtwo−sided,A : n → , for ∅ 6= A
given by
(
−d n\A (x) if x ∈ A,
dtwo−sided,A (x) :=
dA (x)
otherwise.
It is easy to see that computability of the two–sided distance function implies computability of the usual distance function. Note that the usual distance function dA
of a nonempty closed set A ⊆ n and the two–sided distance function dtwo−sided,B
n
of a nonempty closed set B
are total and Lipschitz–continuous. Thus, their
computability can be described using Lemma 5.
The computability notions based on distance functions are compatible with the
classical decidability notion for subsets A ⊆ . If a set A ⊆ is considered a subset
of the real line by embedding
into
then the functions dA and dtwo−sided,A are
in the
identical. They are computable if, and only if, A is a decidable subset of
classical sense. Finally, it is easy to see that the two–sided distance function of the
epigraph of the exponential function 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, it is c.e. and its complement is c.e. as well. The first two statements of Lemma 6
are analogous statements over n. For a proof of the lemma see [15].
Lemma 6. Fix some n ≥ 1.
n
the following two conditions are equivalent.
1. For a nonempty closed set A
(a) The two–sided distance function dtwo−sided,A is computable.
(b) The interior of A is c.e. open, the boundary of A is c.e. closed, and the
complement of A is c.e. open.
2. For a nonempty closed set A ⊆ n the following two conditions are equivalent.
(a) The distance function dA is computable
(b) A is c.e. closed, and the complement of A is c.e. open.
3. For a nonempty closed set A ⊆ n the following two conditions are equivalent.
(a) The restriction dA | n\A of the distance function dA to the complement of A
is computable, and the complement of A is c.e. open.
IS THE MANDELBROT SET COMPUTABLE?
9
(b) The boundary of A is c.e. closed, and the complement of A is c.e. open.
n
2.5. A Hierarchy Result. Consider some non-empty closed subset A
. We
already mentioned that if dtwo−sided,A is computable then also dA is computable. It
is also clear that if dA is computable then also dA | n\A is computable. But are these
really different conditions? One may wonder especially about the difference between
the computability of dA | n\A , the restriction of dA to the complement of A, and the
computability of dA itself. After all, on A the distance function dA has constant value
0 anyway. But indeed, these computability notions are different.
n
Proposition 7. For any n ≥ 1, there exist nonempty closed subsets A, B
c.e. open complement such that
with
• dA is computable, but dtwo−sided,A is not,
• dB | n\B is computable, but dB is not.
Furthermore, for n ≥ 2, there are sets A, B ⊆ n with these properties which are
additionally compact, equal to the closure of their interior, connected, and simply
connected.
Proof of Proposition 7. The statement without the additional topological conditions
is contained in [15, Theorem 3.15]. Therefore, here we consider only n ≥ 2 and
construct suitable sets A, B ⊆ n satisfying also the additional conditions.
Let K ⊆ be an undecidable, c.e. set. Let h :⊆ → be a computable function
with dom(h) = K and such that the set {(i, m) ∈ 2 | i ∈ K and h(i) ≤ m} is
decidable. For each i ∈ K, the number h(i) can be considered as the halting time
of an algorithm halting for all i ∈ K but for no other i ∈ . One can obtain such a
function for example by considering a total, computable function g :⊆ → with
range(g) = K, and by defining h by h(i) := min{n ∈ | g(n) = i} for all i ∈ K. We
n
define nonempty subsets A, B
by
A := [0, 2]n \
[
[0, 2−i ) × (2−i − 2−i−h(i)−2 , 2−i + 2−i−h(i)−2 )n−1
i∈K
B := A \
[
(2−i − 2−i−2 , 2−i + 2−i−2 )n
i∈K
!
!
,
.
The set A is obtained by drilling for each i ∈ K a thin hole into the box [0, 2]n .
The set B is obtained by additionally carving out a cave at the end of each of these
holes. It is clear that each of the sets A and B is compact, equal to the closure of its
interior, connected, and simply connected. We prove that the sets A and B satisfy
also the other claimed conditions.
The function dA is computable. In order to see this, assume that a ρn –name of a
point x ∈ n and a number k are given. Then, we can compute the finite set
I (k) := {i ∈
| i < k and i ∈ K and h(i) < k}
10
PETER HERTLING
and, for

A(k) := [0, 2]n \ 
[
i∈I (k)

[0, 2−i ) × (2−i − 2−i−h(i)−2 , 2−i + 2−i−h(i)−2 )n−1  ,
we can compute dA(k) (x) with arbitrary precision, for example with precision 2−k−1 .
Since |dA (x) − dA(k) (x)| ≤ 2−k−2 , then we know dA (x) with precision 2−k−2 + 2−k−1 ≤
2−k . The function dtwo−sided,A is not computable. Otherwise, the sequence of numbers
(dtwo−sided,A (2−i , . . . , 2−i ))i would be a computable sequence of real numbers because
((2−i , . . . , 2−i ))i is a computable sequence of points in n, and computable functions
map computable sequences to computable sequences. But, we observe
(
=0
if i ∈ K,
dtwo−sided,A (2−i , . . . , 2−i )
−i−2
< −2
if i 6∈ K.
Hence, given i ∈ , by computing a rational 2−i−3 –approximation of the number
dtwo−sided,A (2−i , . . . , 2−i ) and by checking whether this rational approximation is ≥
−2−i−3 or not, we could decide whether i ∈ K or not. But, K was assumed to be
undecidable.
It is clear that the complement n \B of B is c.e. open, and that the boundary of B
is c.e. closed. By Lemma 6, this implies that the function dB | n\B (x) is computable.
But the function dB is not computable. Otherwise, the sequence (dB (2−i , . . . , 2−i ))i
would be computable. But, we observe
(
2−i−2 if i ∈ K,
dB (2−i , . . . , 2−i ) =
0
if i 6∈ K.
As above, given i ∈ , we could decide whether i ∈ K or not by using this computable
sequence. But, K was assumed to be undecidable.
2.6. Computable Riemann Mappings. For computability questions we always
identify the complex plane and the real plane. In this subsection we formulate a
corollary of an effective version of the Riemann mapping theorem.
Let K ⊆ be a compact, connected, and simply connected subset of the complex
plane containing more than one point. Then there are a unique real number ρ > 0
and a unique conformal mapping Φ of the complement of K onto the complement
{z ∈ | |z| > ρ} of the closed disk with radius ρ such that Φ has a simple pole at
infinity and its Laurent series starts with z. This is a well known corollary of the
Riemann mapping theorem; see Henrici [14, Corollary 5.10d].
Proposition 8. If the complement of K is c.e. open and the boundary of K is c.e.
closed, then the number ρ, the mapping Φ, and its inverse Φ−1 are computable.
Proof. We can derive this statement from the effective version of the Riemann mapping theorem formulated in [15] by using the same arguments as those [14, p. 381]
used for deriving the corollary, mentioned above, of the Riemann mapping theorem.
Note also that the inverse of a computable conformal mapping on a c.e. open set is
again a computable conformal mapping; see [15, Theorem 4.5].
IS THE MANDELBROT SET COMPUTABLE?
11
Since the boundary of K is c.e. closed, we can fix some computable point z0 ∈ K.
1
, is computable.
The mapping h1, defined in \ {z0} and given by h1 (z) := z−z
0
Its inverse is computable as well. Certainly, h1( \ K) ∪ {0} is c.e. open, simply
connected, a proper subset of , and the boundary of h1 ( \ K) ∪ {0} is c.e. closed.
By the effective version of the Riemann mapping theorem in [15], there is a (unique)
computable, conformal mapping f : B(0, 1) → h1( \ K) ∪ {0} with f (0) = 0 and
f 0 (0) > 0. The number ρ := 1/f 0 (0) is a computable real number, and the inverse
function f −1 of f is a computable function; see e.g. [15, Prop. 4.1 and Theorem 4.5].
ρ
| |z| > ρ}, is a computable
The function Ψ := h−1
1 ◦ f ◦ (z 7→ z ), defined on {z ∈
conformal mapping from {z ∈
| |z| > ρ} onto \ K. Its inverse Φ := Ψ−1 is
computable as well, has a simple pole at infinity, and its Laurent expansion starts
with z.
3. The Mandelbrot Set
In this section we state our results and some conjectures with respect to computability of the Mandelbrot set. First, we present three results. Then, we formulate
two well-known conjectures about the Mandelbrot set and two conjectures concerning computability of the Mandelbrot set. In the third subsection we discuss the last
conjecture in more detail. In the fourth subsection we show that one of the earlier
computability statements can be strengthened.
3.1. Computability of Sets Related to the Mandelbrot Set. For an introduction to the Mandelbrot set see Branner [5]. We remind the reader of some basic
properties already mentioned in the introduction. For any c ∈ the sequence p◦k
c (0)
either tends to infinity for k tending to infinity or stays within the closed disk of radius
2 around the origin. The Mandelbrot set is the set of all complex parameters c such
that the sequence (p◦k
c (0))k is bounded. The Mandelbrot set is a compact, connected,
and simply connected subset of the complex plane, contained in the closed disk of
radius 2 around the origin. Furthermore, it is equal to the closure of its interior. For
its role in connection with Julia sets see e.g. Branner [5].
We are interested in computability properties of the Mandelbrot set. For that
purpose we always identify the complex plane and the real plane. The following
observation [33, Exercise 5.1.32] is well–known.
Proposition 9. The complement
\ M of the Mandelbrot set is c.e. open.
Proof. According to Lemma 1 it is sufficient to construct an algorithm which, given
a ρ2 –name of a complex number c, stops if, and only if, c 6∈ M . Remember that a
ρ2 –name of c is essentially an infinite sequence of pairs (am , bm ) of rational numbers
with |am + ibm − c| ≤ 2−m for all m. It is clear that there exists an algorithm (Turing
machine) with the following properties: given as input a ρ2 –name of a complex
number c and an integer n, the algorithm computes a rational number q (n) with
|q (n) − p◦n
c (0)| < 1. Using this algorithm we can construct an algorithm with the
following property. Given as input a ρ2 –name of a complex number c, it does the
following, starting with n = 0: it computes the number q (n) , checks whether q (n) is
larger than 3, and stops if this is case. Otherwise it continues with n + 1. It is clear
that this algorithm stops if, and only if, c 6∈ M . Hence, \ M is c.e. open.
12
PETER HERTLING
Unfortunately we do not know whether also the interior of M is c.e. open. But at
least we can show that a certain important subset of the interior of M is c.e. open
and that the boundary of M is c.e. closed. In order to formulate and prove these
statements we introduce some more terminology. Consider a complex number z0 ∈
and a complex polynomial p.
• z0 is a periodic point for p if there is some integer k > 0 such that p◦k (z0) = z0.
The smallest k > 0 with this property is called the period of z0 .
• z0 is a preperiodic point for p if there are some k > 0 and l ∈
such that
p◦(k+l) (z0 ) = p◦l (z0). The smallest k > 0 such that there is some l with this
property is called the period of z.
• z0 is a strictly preperiodic point for p if it is preperiodic for p but not periodic
for p.
Now assume that z0 is periodic for p with period k > 0. Then the set of points
z0, z1, . . . , zk−1 where
zj := p◦j (z0 )
is called a cycle. The derivative of p◦k at z0 is called the multiplier of the cycle. This
makes sense because by the chain rule
(1)
(p◦k )0(z0 ) =
k−1
Y
p0 (zj ) = (p◦k )0 (zi)
j=0
for all i ∈ {0, . . . , k − 1}. A cycle is called attracting if the absolute value of its
multiplier is smaller than 1. We define the set
H(M ) := {c ∈
| pc has an attracting cycle}.
It is well known that H(M ) is an open subset of the interior of the Mandelbrot set.
Its connected components are called the hyperbolic components of the Mandelbrot
set. Thus, H(M ) is the union of the hyperbolic components of the Mandelbrot set.
Later we shall prove a strengthening of the following observation.
Proposition 10. The union of the hyperbolic components of the Mandelbrot set is
c.e. open.
Proof. According to Lemma 1 it is sufficient to construct an algorithm which, given a
ρ2 –name of a complex number c, that is, given a fast converging sequence of rational
approximations for c, stops if, and only if, pc has an attracting cycle. Such an
algorithm could work as follows in stages 0, 1, 2, . . . . In stage hk, mi, for each fixed
−m
0
point z0 of p◦k
–approximation qz0 ,m of (p◦k
c , it computes a rational 2
c ) (z0 ) (note
that using an effective version of the Fundamental Theorem of Algebra (Specker [30])
◦k
one can compute the fixed points of p◦k
c , that is, the zeros of pc (z) − z, with any
required precision), and checks whether the absolute value of this rational complex
number is smaller than 1 − 2−m . If the answer is yes for at least one fixed point z0 of
p◦k
c , then the algorithm stops. Otherwise it enters the next stage hk, mi + 1.
It is clear that this algorithm stops on input c if, and only if, pc has an attracting
cycle.
Proposition 11. The boundary ∂M of the Mandelbrot set is c.e. closed.
IS THE MANDELBROT SET COMPUTABLE?
13
Proof. It is well known (see e.g. Branner [5]) that H(M ) is a subset of the interior
of M and that the boundary of M is contained in the closure of H(M ). Thus, the
boundary ∂M of M is equal to the intersection of the closure of \ M and the closure
of H(M ). The boundary ∂M of M is equal to the boundary of the closure of \ M .
The assertion follows from Propositions 9 and 10 and from Lemmata 2 and 3.
Remark. A complex number c ∈
is called a Misiurewicz point if 0 is a strictly
preperiodic point for the polynomial pc . It is well known (see e.g. Branner [5]) that
the set of Misiurewicz points is a dense subset of the boundary. One can construct
a computable sequence of complex numbers which contains exactly the Misiurewicz
points. This gives another proof for the c.e. closedness of the boundary of M ; compare
Lemma 1. This was observed independently from the author also by Olha Shkaravska
(email May 29, 2000).
Corollary 12. The restriction dM | \M of the distance function of the Mandelbrot
set to the complement of the Mandelbrot set is computable.
Proof. By Propositions 9 and 11, and by Lemma 6.
Thus, there is an algorithm which for any point c outside of M computes dM (c)
with arbitrary precision.
3.2. Four Conjectures about the Mandelbrot Set. A famous conjecture about
the Mandelbrot set is the following.
Conjecture 1. The Mandelbrot set is locally connected.
Douady and Hubbard [9] have shown that if this conjecture is true, then also the
following famous conjecture, known under the name hyperbolicity conjecture, is true.
Conjecture 2. The interior of the Mandelbrot set is equal to the union of its hyperbolic components.
This conjecture implies computability of the Mandelbrot set in our setting.
Theorem 13. If the hyperbolicity conjecture is true, then the two–sided distance
function dtwo−sided,M of the Mandelbrot set is computable.
Proof. By Propositions 9, 10, and 11, and by Lemma 6.
Therefore, we formulate:
Conjecture 3. The two–sided distance function dtwo−sided,M of the Mandelbrot set is
computable.
By Propositions 9 and 11 and Lemma 6, this conjecture is equivalent to the conjecture that the interior of the Mandelbrot set is c.e. open. It implies the following
conjecture which, by Proposition 9 and Lemma 6, is equivalent to the conjecture that
the Mandelbrot set is c.e. closed.
Conjecture 4. The distance function dM of the Mandelbrot set is computable.
14
PETER HERTLING
Note that the facts that the complement of the Mandelbrot set is c.e. open and the
boundary is c.e. closed imply that the the restriction dM | \M of the distance function
dM to the complement of the Mandelbrot set is computable (Corollary 12). One might
wonder whether perhaps these facts in combination with other topological properties
of M might immediately imply that the distance function of M is computable. For
example, we already mentioned that the Mandelbrot set is compact. Furthermore it
is known that M is the closure of its interior and that it is connected and simply
connected and, therefore, its complement is connected as well. But we have already
seen in Proposition 7 that there is a subset of 2 which has all these properties but
which does not have a computable distance function. In Proposition 7 we have also
seen that there is a compact, simply connected subset of 2 which is equal to the
closure of its interior and whose distance function is computable, but whose two–sided
distance function is not computable.
3.3. The Distance Function, the Function Φ, and the Escape Time. Conjecture 4 can be reformulated using the escape time, which is, for a given point c outside
M , the smallest number k such that |p◦k
c (0)| is larger than some fixed large bound.
Lemma 14. The distance function dM is computable if, and only if, there is a total,
computable, function e : → such that for k ∈ and c ∈ ,
(2)
◦e(k)
if |pc
(0)| < 3 then dM (c) < 2−k .
Proof. First, let us assume that there is such a function e. We have to show that, given
a ρ2 –name of a complex number c and a number k ∈ , we can compute a rational
2−k –approximation of dM (c). Therefore, we compute a rational 1/4–approximation
◦e(k)
q of pc (0). If q > 5/2, then we know c ∈ \ M , and we can invoke an algorithm
for dM | \M (remember that according to Corollary 12 the function dM | \M is computable) in order to compute dM (c) = dM | \M (c) with precision 2−k . If q ≤ 5/2,
◦e(k)
then we know that |pc (0)| < 3, hence dM (c) < 2−k . Then 0 is a rational 2−k –
approximation of dM (c).
For the other direction, at first we note that we can enumerate a list ((im , jm ))m of
2
m
pairs (im , jm ) ofSintegers such that |p◦j
c (0)| > 3 for all c ∈ B (im ), for all m, and such
2
that the union m∈ B (im ) of all these balls covers \ M . Let us assume that dM is
computable. Then, given k, we can in parallel also enumerate a list of rational balls
−k
covering exactly the set d−1
M ([0, 2 )). The union of the rational balls in both lists
together certainly covers . Hence, after finitely many steps we will have covered the
compact ball {c ∈ | |c| ≤ 3}. We define e(k) to be the maximum of all numbers
jm which have appeared in the first list so far. It is clear that the function e defined
in this way is computable. We show that it satisfies (2). It is clear that e(k) ≥ 1 for
all k. For |c| > 3 we observe that |p◦l
c (0)| > 3 for all l ≥ 1. And for c with |c| ≤ 3
and some w with |w| ≥ 3,
|pc (w)| = |w2 + c| ≥ 9 − 3 = 6.
0
◦l
0
Thus, if |c| ≤ 3 and |p◦l
c (0)| ≥ 3 for some l, then |pc (0)| > 3 for all l > l. We
conclude that e satisfies (2).
IS THE MANDELBROT SET COMPUTABLE?
Let the function ê :
→
ê(k) := min{l ∈
15
be defined by
| |p◦l
c (0)| ≥ 3 for all c ∈
with dM (c) ≥ 2−k }
for all k ∈ . This function is well defined because for all c with |c| > 3 we have
|pc (0)| = |c| > 3 and because for every k ∈ the set {c ∈ | 2−k ≤ dM (c) and |c| ≤
3} is a compact subset of \ M . Furthermore, from the final arguments in the
proof of Lemma 14 it is clear that a function e :
→
satisfies (2) if, and only
if, it satisfies e(k) ≥ ê(k) for all k ∈ . Thus, the question whether the distance
function dM is computable is equivalent to the question whether the function ê can
be bounded from above by a computable function.
Carleson and Gamelin [8, Chapter VIII, Theorem 4.2] have shown that the Mandelbrot set P
is even locally connected if the function ê satisfies the following growth
∞
−k
condition:
< ∞. Through the list of Conjectures 1 to 4 we already
k=0 ê(k) · 2
know that dM is computable if the Mandelbrot is locally connected, and, therefore,
also if ê satisfies this growth condition. But it can also be seen directly that this
growth
for ê implies computability of dM : if N is a natural number larger
Pcondition
∞
than k=0 ê(k) · 2−k , then the function e0 :
→
defined by e0(k) := N · 2k is
0
computable and satisfies e (k) ≥ ê(k) for all k, hence, (2).
It is also interesting to note that there is a close connection between the escape time
and the potential function of M . Since the Mandelbrot set is compact and simply
connected, there exists a unique number ρ > 0 and a unique function Φ mapping the
complement of the Mandelbrot set conformally onto the complement of the disk with
radius ρ and having a simple pole at infinity and a Laurent expansion starting with
z; compare Proposition 8. In fact, for the Mandelbrot set it is known that ρ = 1.
This function Φ was explicitly constructed by Douady and Hubbard [9] when they
proved that M is connected.
Proposition 15. The function Φ and its 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 Φ0 resp. on the potential function
G(c) := log(|Φ(c)|) of M (where log denotes the natural logarithm) and its derivative
G0 have been very useful for actual high precision computations of the Mandelbrot
set; compare Peitgen, Saupe [22]. This is based on the fact that, using the Schwarz
Lemma and the Koebe 41 Theorem, one can estimate the distance of a point outside
M from M in terms of Φ and its derivative Φ0 resp. in terms of G and G0 :
(3)
2 · sinh G(c)
sinh G(c)
< dM (c) <
G(c)
0
2·e
· |G (c)|
|G0 (c)|
for all c ∈ \ M ; compare Milnor [20], Fisher [11]. Using a recursion scheme one
can compute G(c) and G0 (c) even quite fast. This is due to the fact that G(c) =
(0)|)/2n ) for all c ∈ \ M .
limn→∞ (log(|p◦n+1
c
Finally, the function
ee(c) := − log2 G(c),
16
PETER HERTLING
for c ∈ \ M is a “smoothed out approximation” of the escape time; see Milnor [20].
That means the following. For R ≥ 3 and c ∈ \ M with |c| ≤ 3 we define
eR(c) := min{k ∈
| |p◦k
c (0)| ≥ R}.
Then there is a constant C, depending on R, but not on c, such that |e
e(c)−eR (c)| ≤ C
for all c ∈ \ M with |c| ≤ 3. For completeness sake we prove this, even a slightly
stronger statement; compare Milnor [20].
Lemma 16. For all c ∈
n∈ ,
\ M with |c| ≤ 3 and R ≥ 3, and with zn := pnc (0) for all
|e
e(c) − eR(c) + 1 − log2 (log |zeR (c) |)| <
Proof. Fix some R ≥ 3 and some c ∈
using m := n − eR (c),
R2
12
.
log R
\ M with |c| ≤ 3. We obtain for all n ≥ eR (c),
log |zn+1 |
2n
m+1
= 2−eR (c)+1 · log |zeR (c)+m+1 |1/2
!
1/2
1/2m+1
|z
|
|z
|
e
(c)+1
e
(c)+m+1
= 2−eR (c)+1 · log |zeR (c) | · R
· ...· R
|zeR (c) |
|zeR (c)+m |1/2m

1/2
1/2m+1 
c c
−eR (c)+1


= 2
· log |zeR (c)| · 1 + 2 · . . . · 1 + 2
zeR (c) zeR (c)+m !
c
c
1
1
−eR (c)+1
= 2
· log |zeR (c) | + · log 1 + 2 + . . . + m+1 · log 1 + 2
.
2
zeR (c) 2
zeR (c)+m Due to G(c) = limn→∞ (log(|zn+1 |)/2n ) we obtain
3 3
9
|c|
eR (c)−1
,
|G(c) · 2
− log |zeR (c) || ≤ − log 1 − 2 ≤ · 2 =
R
2 R
2R2
where we used |zeR (c)| ≥ R, |c| ≤ 3, R ≥ 3, and log0 (x) ≤ 3/2 for all x ≥ 2/3. Finally,
we obtain
9
9
0
|e
e(c) − eR(c) + 1 − log2 (log |zeR (c) |)| ≤ log2 log |zeR (c)| −
·
2
2R
2R2
9
≤ log02 (log(R) − 1/2) ·
2R2 9
log(3) − 1/2
· log(R) ·
≤ log02
log(3)
2R2
12
<
2
R log R
IS THE MANDELBROT SET COMPUTABLE?
17
Note that R ≤ |zeR (c) | ≤ R2 + 3 for c, R, and zeR (c) as in the lemma.
This relation between ee and the escape time eR is interesting because Equation
(3) tells us that for every ε > 0 there is some δ > 0 such that the following is true:
if 0 < dM (c) < δ, then the distance dM (c) is up to a factor smaller than 2 + ε and
larger than 1/(2 + ε) equal to
G(c)
1
=
.
0
|G (c)|
log(2) · |e
e0(c)|
Thus, for c close to M but not inside M , the distance dM (c) is proportional to the
inverse of the absolute value of the derivative of the escape time.
3.4. Computability of the Hyperbolic Components. Finally, we show that
at least with respect to the hyperbolic components one can compute more than
stated in Proposition 10. We call a sequence (Uk )k of open subsets of n uniformly
c.e. open if the set {hi, ki | closure(Bn (i)) ⊆ Uk } is computably enumerable. And
we call a sequence (Ak )k of closed subsets of n uniformly c.e. closed if the set
{hi, ki | Bn (i) ∩ Ak 6= ∅} is computably enumerable.
Proposition 17. The hyperbolic components of M can be ordered into a sequence
(Wi )i such that this sequence is uniformly c.e. open, the sequence (∂Wi)i of the
boundaries of the components Wi is uniformly c.e. closed, and the sequence ( \
closure(Wi ))i of the complements of the closures of the Wi is uniformly c.e. open.
Using [15, Theorem 3.14] one sees that, similarly to the first statement in Lemma 6,
this means that the function
×
→
(i, c) → dtwo−sided,closure(Wi ) (c)
is computable, where computability of a Lipschitz continuous function from
to can be characterized in a way similar to Lemma 5.
×
n
Proof of Proposition 17. Remember that H(M ) is the set of all c such that the quadratic polynomial pc has an attracting cycle. The set H(M ) is open, due to the
Implicit Function Theorem. It is known that a quadratic polynomial can have at
most one attracting cycle. Therefore, for all c within one connected component of
H(M ) the period of the unique attracting cycle of pc is the same. It is also known
that each hyperbolic component, i.e., each connectedness component of H(M ), is an
open, simply connected subset of the interior of M . Even more, for each hyperbolic
component W , the function ρW which maps c ∈ W to the multiplier of the unique
attracting cycle of pc is a conformal mapping of W onto the open unit disk D. This
implies that each component W of H(M ) contains exactly one point c with multiplier
zero. These values c are called the centers of the hyperbolic components. Due to the
fact that the multiplier of a periodic cycle z0, . . . , zk−1 of pc is given by
(4)
0
p◦k
c
(z0) =
k−1
Y
j=0
p0c (zj )
k
=2 ·
k−1
Y
j=0
zj ,
18
PETER HERTLING
the centers of the hyperbolic components are exactly the values c such that 0 is a
periodic point, i.e., the solutions c of the equations
(5)
p◦k
c (0) = 0
for k = 1, 2, 3, . . . . Note that p◦k
c (0) is a polynomial in c. It is known that the roots
of these polynomials are simple; for a short proof, due to Gleason, see [10, p. 108].
◦l
Of course, solutions of p◦k
c (0) = 0 are also solutions of pc (0) = 0 if k divides l. But
◦l
then, i.e., if k divides l, the polynomial p◦k
c divides the polynomial pc . Thus, using
an effective version of the fundamental theorem of algebra (Specker [30]) one can
compute a list (ci )i containing all solutions c of (5) for all k ≥ 1 without repetitions.
This list is a list of all the centers of the hyperbolic components without repetitions.
Let Wi be the hyperbolic component with center ci .
We wish to show that the sequence (Wi )i has the properties stated in the proposition. Let us assume that an index 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 covering exactly H(M )(k) . The closure of H(M )(k) consists exactly of all c such
that pc has a non-repelling cycle (that means, the absolute value of the multiplier
is less than or equal to 1) of period k; see [21] or [27]. Thus, \ closure(H(M )(k) )
consists of all those parameters c such that all periodic points of period k of pc have
multiplier with absolute value greater than 1. Again, by a similar reasoning as in the
proof of Proposition 10 one can see that one can enumerate a list L2 of rational balls
covering \closure(H(M )(k) ). It is also known that the closures of the components of
H(M )(k) are compact, pairwise disjoint sets; see [21] or [27]. Hence, using the list L2,
one can compute a simply closed piecewise linear curve C with finitely many rational
breakpoints lying inside the area covered by the balls in the second list L2 such that
ci lies inside the area encircled by this curve —let us call this (open) area A— while
all other centers of components of H(M )(k) lie outside. Then, Wi = H(M )(k) ∩ A
while the other hyperbolic components with the same period k lie in \ closure(A).
We can also compute a list L3 of rational balls covering A and a list L4 of rational
balls covering \ closure(A). Using the lists L1 and L3 it is clear that we can also
compute a list of rational balls covering exactly Wi = H(M )(k) ∩ A. Since all this can
be done uniformly in i, the list (Wi )i is uniformly c.e. open. And by taking the union
of the lists L2 and L4 we obtain a list of rational balls covering exactly \ Wi . Thus,
the list ( \ closure(Wi ))i is uniformly c.e. open as well. Finally, the boundary of Wi
is equal to the intersection of the closure of Wi and the closure of \ closure(Wi ).
By uniform versions of Lemma 2 and Lemma 3 it follows that the sequence (∂Wi )i
is uniformly c.e. closed.
In the proof of Proposition 17 we mentioned that for any i, the period of the
unique attracting cycle of pc for all c ∈ Wi is constant. Thus, one can speak of the
period of a hyperbolic component. It is clear that one can compute a list (Wi )i as in
Proposition 17 such that these periods are non-decreasing for increasing i.
IS THE MANDELBROT SET COMPUTABLE?
19
Since the union of a uniformly c.e. open sequence of sets is obviously a c.e. open set,
Proposition 17 gives another proof of Proposition 10. In addition, one can deduce that
the boundary of H(M ) is c.e. closed. But this is clear anyway from Proposition 11
because the boundary of H(M ) is equal to the boundary of M : on the one hand,
the boundary of M is contained in the closure of H(M ) (see Branner [5]), on the
other hand, the boundary of any hyperbolic component is contained in the boundary
of M (see Schleicher [27]), and the boundary of H(M ) is equal to the closure of the
union of the boundaries of all hyperbolic components. If the hyperbolicity conjecture
(Conjecture 2) is true, then the two–sided distance function of closure(H(M )) is
equal to the two–sided distance function of M and computable. Since H(M ) is
a bounded set, the supremum of the radii of disks that can be inscribed into a
hyperbolic component of period k must tend to zero for k tending to infinity. It
would be interesting to have an effective upper bound for this supremum, tending to
zero for k tending to infinity. Then one could compute at least the distance function
d \H(M ) of the complement of the union of the hyperbolic components.
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Peter Hertling, Theoretische Informatik, FernUniversität Hagen, 58084 Hagen,
Germany
E-mail address: [email protected]