Non computable Mandelbrot-like set for a one-parameter complex family

TL;DR

This paper demonstrates that for certain computable complex parameters, the bifurcation locus of a specific one-parameter complex family is not Turing computable, revealing limits of algorithmic predictability in complex dynamics.

Contribution

It establishes the existence of computable parameters where the bifurcation locus is non-Turing computable, highlighting fundamental limits in computational complex dynamics.

Findings

Existence of non-Turing computable bifurcation loci for certain parameters

Identification of computable complex numbers with non-computable bifurcation sets

Advancement in understanding the computational complexity of dynamical systems

Abstract

We show the existence of computable complex numbers $\lambda$ for which the bifurcation locus of the one parameter complex family $f_{b}(z) = \lambda z + b z^{2} + z^{3}$ is not Turing computable.

Equations

$$f_b=\lambda z+b{z}^2+z^3,b\in \mathbb{C},$$

$${\mathrm{dist}}_H(K_1,K_2)=\underset{ϵ}{\inf }\{K_1\subset U_{ϵ}(K_2)\text{ and }{K}_2\subset U_{ϵ}(K_1)\},$$

$${\mathrm{dist}}_H(C_n,K)<2^{-n}.$$

$$\underset{\delta \to 0}{\lim }m(\delta )=0\text{ and }|\varphi (x)-\varphi (y)|<m(|x-y|)$$

$$G_f(z)=\underset{n\to +\infty }{\lim }\frac{1}{d^n}\log |f^n(z)|>0.$$

$${\phi }_f:\hat{\mathbb{C}}\setminus \overline{\mathbb{D}}⟶\hat{\mathbb{C}}\setminus K(f)\text{ with }{\phi }_f(\infty )=\infty ,\text{ and }{\phi }_f^{\prime }(\infty )=1$$

$$U_f:=\{z\in \mathbb{C}\mid G_f(z)>G_f(\omega )\}$$

$${\phi }_f:U_f\to \{z\in \hat{\mathbb{C}}\mid |z|>\exp (G_f(\omega ))\},$$

$$\{{\phi }_f^{-1}(r\exp (2\pi it))\mid \exp (G_f(0))<r<+\infty \}.$$

$$f_b(z)=z+b{z}^2+z^3$$

$$\begin{array}[]{cccl}\Phi_{\lambda}:&\mathbb{C}\setminus\mathcal{M}_{\lambda}&\to&\mathbb{C}\setminus\overline{\mathbb{D}}\\ &b&\mapsto&\Phi_{\lambda}(b)=\varphi_{b}(\omega^{\prime}_{2}).\end{array}$$

$${\mathcal{E}}_{\lambda }(v):={\Phi }_{\lambda }^{-1}(\{z\in \mathbb{C}\mid |z|=v\}).$$

$${\mathcal{R}}_{\lambda }(t):={\Phi }_{\lambda }^{-1}(\{r\exp (2\pi it)\mid r>1\}).$$

$$|\varphi (x)-\varphi (y)|\le H|x-y{|}^{\alpha }.$$

$$X_{b_1}=\bigcup _{\theta \in \Theta }{R}_{f_{b_1}}(\theta )$$

$$\mathbb{C}\setminus (\overline{{\mathcal{R}}_{\lambda }(1/6)}\cup \overline{{\mathcal{R}}_{\lambda }(1/3)})$$

$$\frac{2p+1}{2^k}=\sum _{i=1}^k\frac{{\epsilon }_i}{2^i},$$

$${\vartheta }^{-}=\sum _{i=1}^k\frac{{\epsilon }_i+1}{3”},\text{ and }{\vartheta }^{+}={\vartheta }^{-}+\frac{1}{2\cdot 3^k}.$$

$$f_{b_{\vartheta }}^{k+1}({\omega }_2(b_{\vartheta }))=\beta (b_{\vartheta }).$$

$$\mathbb{C}\setminus (\overline{{\mathcal{R}}_{\lambda }({\vartheta }^{-}/3)}\cup \overline{{\mathcal{R}}_{\lambda }({\vartheta }^{+}/3)})$$

$${\mathcal{X}}_{\vartheta }=\bigcup _{\theta \in \Theta }{\mathcal{R}}_{\lambda }\left(\frac{{\vartheta }^{-}}{3}+\frac{\theta }{3^{k+1}}\right),$$

$$H_{\vartheta }(b)=h_b^{-1}(f_b^{k+1}({\omega }_2(b)))$$

$$f(z)=z^d+a_{d-1}{z}^{d-1}+\cdots +a_{1}z+a_0$$

$$V_f=\hat{\mathbb{C}}\setminus (K_f\cup S_f),$$

$${\phi }_f(z)=z\cdot \prod _{n=0}^{+\infty }{\left(1+\frac{f(f^n(z))-(f^n(z){)}^d}{(f^n(z){)}^d}\right)}^{d^{-(n+1)}}.$$

$$\frac{|f(f^n(z))-(f^n(z){)}^d|}{|f^n(z){|}^d}\le \frac{1}{4},$$

$$a_n=\frac{f(f^n(z))-(f^n(z){)}^d}{(f^n(z){)}^d}.$$

$$\log \frac{3}{4}\le \log |1+a_n|\le \log \frac{5}{4}.$$

$$\left|\prod _{n=0}^{k-1}{\left(1+a_n\right)}^{d^{-(n+1)}}\right|\le {\frac{5}{4}}^{\frac{1}{d-1}},$$

$${\frac{3}{4}}^{\frac{1}{d^k(d-1)}}\le \left|\prod _{n=k}^{+\infty }{\left(1+a_n\right)}^{d^{-(n+1)}}\right|\le {\frac{5}{4}}^{\frac{1}{d^k(d-1)}}.$$

Full text

D.C. and C.R were partially supported by project DI-782-15/R Universidad Andrés Bello and Basal PFB-03 CMM-Universidad de Chile. M.Y. was partially supported by NSERC Discovery grant.

1. Introduction

For a complex quadratic map $P_{c}(z)=z^{2}+c$, recall that the filled Julia set $K(P_{c})$ corresponds to the set of points $z\in{\mathbb{C}}$ whose orbit under iterations by $P_{c}$ remains bounded, and that the Julia set $J(P_{c})$ is defined as the boundary of $K(P_{c})$. Let us also recall that the Mandelbrot set $\mathcal{M}$ is defined to be the connectedness locus of the family $P_{c}(z)=z^{2}+c,\,\,c\in{\mathbb{C}}$: the set of complex parameters $c$ for which the Julia set $J(P_{c})$ is connected. The boundary of $\mathcal{M}$ corresponds to the parameters near which the geometry of the Julia set undergoes a dramatic change. For this reason, its boundary $\partial\mathcal{M}$ is referred to as the bifurcation locus. The Mandelbrot set is widely known for the spectacular beauty of its fractal structure, and an enormous amount of effort has been made in order to understand its topological and geometrical properties. This effort has greatly relied on computer simulations, and it is most natural to ask whether these simulations can be trusted. A form of this question was first asked by Penrose in [14] and has been a subject of much interest.

The central open conjecture in complex dynamics is known as Density of Hyperbolicity Conjecture. This conjecture is widely expected to be true, and postulates that ${\cal M}$ is the closure of the open set of parameter values $c$ for which $P_{c}$ exhibits hyperbolic dynamics. The latter simply means that $|DP^{n}_{c}|>1$ on a neighborhood of $J(P_{c})$ for some $n\in{\mathbb{N}}$. In [10], Hertling demonstrated that Density of Hyperbolicity Conjecture implies that Mandelbrot set ${\cal M}$ as well as its boundary, the bifurcation locus $\partial\mathcal{M}$, are rigorously computable.

In this paper we show that such a computability property cannot be taken for granted. We consider a different one-parameter family of complex dynamical systems, studied by X. Buff and C. Henriksen in [7]:

[TABLE]

where $\lambda=e^{2i\pi\theta}\in\mathbb{C}$ satisfies $|\lambda|=1$. We denote by $\mathcal{M}_{\lambda}$ the connectedness locus of the family, that is, the set of complex parameters $b$ for which the Julia set $J(f_{b})$ is connected.

Our main result is the following.

Main Theorem. There exists a computable (by an explicit algorithm) value of $\lambda$ such that the bifurcation locus $\partial\mathcal{M}_{\lambda}$ is not computable.

A principal result of [7] is that for each $\lambda$ of modulus 1, the bifurcation locus $\partial\mathcal{M}_{\lambda}$ contains quasi-conformal copies of the quadratic Julia set $J(\lambda z+z^{2})$ (see Figure 1 for an illustration). The proof of the Main Theorem relies on a computable version of this statement, which is our principal technical result.

2. Preliminaries on Computability

2.1. Rudiments of Computable Analysis and applications to Julia sets

We give a very brief summary of relevant notions of Computability Theory and Computable Analysis. For a more in-depth introduction, the reader is referred to [17, 6]. As is standard in Computer Science, we formalize the notion of an algorithm as a Turing Machine [16].

We will call a function $f:{\mathbb{N}}\to{\mathbb{N}}$ computable (or recursive), if there exists an algorithm ${\cal A}$ which, upon input $n$, outputs $f(n)$.

Extending algorithmic notions to functions of real numbers was pioneered by Banach and Mazur [2, 12], and is now known under the name of Computable Analysis. Let us begin by giving the modern definition of the notion of computable real number, which goes back to the seminal paper of Turing [16]. By identifying $\mathbb{Q}$ with $\mathbb{N}$ through some effective enumeration, we can assume algorithms can operate on $\mathbb{Q}$.

Definition 2.1**.**

A real number $x$ is called computable if there is a computable function $f:{\mathbb{N}}\to{\mathbb{Q}}$ such that

$|f(n)-x|<2^{-n};$

Algebraic numbers or the familiar constants such as $\pi$, $e$, or the Feigembaum constnt [11] are all computable. However, the set of all computable numbers ${\mathbb{R}}_{C}$ is necessarily countable, as there are only countably many computable functions.

For more general objects, computability is typically defined according to the following principle: object $x$ is computable if there exists an algorithm $\mathcal{A}$ which, upon input $n$, outputs a finite suitable description of $x$ at precision $2^{-n}$. In this case we say that algorithm $\mathcal{A}$ computes object $x$.

For instance, computability of compact subsets of ${\mathbb{R}}^{\ell}$ is defined as follows. Recall that Hausdorff distance between two compact sets $K_{1}$, $K_{2}$ is

[TABLE]

where $U_{\epsilon}(K)=\bigcup_{z\in K}B(z,\epsilon)$ stands for an $\epsilon$-neighbourhood of a set.

We say that $K\Subset{\mathbb{R}}^{\ell}$ is computable if there exists an algorithm $A$ which, upon input $n\in{\mathbb{N}}$, outputs a finite set $C_{n}\subset\mathbb{Q}^{\ell}$ of points with rational coordinates such that

[TABLE]

An equivalent, and more intuitive, way of defining a computable set is the following. Let us say that a pixel is a dyadic cube with side $2^{-n}$ and dyadic rational vertices. A set $K$ is computable if there exists an algorithm $\mathcal{A}$ which given a pixel with side $2^{-n}$ outputs [math] if the center of the pixel is at least $2\cdot 2^{-n}$-far from $K$, outputs $1$ if the center is at most $2^{-n}$-far from $K$, and outputs either [math] or $1$ in the “borderline” case. In other words, we can visualise $K$ on a computer screen and zoom-in with arbitrarily high magnification.

In this paper we will speak of uniform computability whenever a group of computable objects (functions, sets, etc) is computed by a single algorithm:

the objects $\{x_{\gamma}\}_{\gamma\in\Gamma}$ are computable uniformly on a countable set $\Gamma$ if there exists an algorithm ${\cal A}$ with an input $\gamma\in\Gamma$, such that for all $\gamma\in\Gamma$, ${\cal A}_{\gamma}:={\cal A}(\gamma,\cdot)$ computes $x_{\gamma}$.

For instance, a sequence $x_{n}$ of computable points is uniformly computable if there is a single algorithm $\mathcal{A}$ which for every $n$ and $m$ outputs a rational number satisfying $|\mathcal{A}(n,m)-x_{n}|<2^{-m}$.

Open sets can be described by means of rational balls: balls with rational centres and radii. An open set $A\subset{\mathbb{R}}$ is called lower-computable or recursively enumerable (r.e.) if it is the union of a computable sequence of rational balls. A function $f$ is a computable function on some set $S\subset{\mathbb{R}}$ if the preimages of rational balls are uniformly lower-computable open (in $S$) sets. That is, if there are uniformly lower-computable open sets $U_{n}$ such that $f^{-1}(B_{n})=U_{n}\cap S$, where $(B_{n})_{n}$ is an enumeration of all the rational balls. It can be verified that this definition of computability for a function $f$ is equivalent to being able to compute $f$ in the following sense: given an arbitrarily good approximation of the input of $f$ in $S$, it is possible to algorithmically approximate the value of $f$ with any desired precision. Computability of functions and open sets of ${\mathbb{C}}$, ${\mathbb{R}}^{n}$, etc…, is defined in a similar fashion. We refer to [17].

We will use the following terminology. A compact set $K$ is lower computable if there is a sequence $x_{n}\in K$ of uniformly computable points which is dense in $K$. It is called upper-computable if its complement is a lower-computable open set.

Example 2.1**.**

The filled Julia set $K(P)$ of a computable polynomial $P$ on $\mathbb{C}$ is always upper computable. For, let $B$ be a closed rational ball containing $K$. Then, $\mathbb{C}\setminus K=\bigcup_{n\in\mathbb{N}}P^{-n}(\mathbb{C}\setminus B)$ which, since $\mathbb{C}\setminus B$ is a recursively enumerable open set and $P$ is computable, is an upper computable set.

Example 2.2**.**

The Julia set $J(P)$ is always a lower computable set. Indeed, it is not hard to see that the set of repelling periodic points of a computable polynomial $P$ can be algorithmically enumerated (periodic points are uniformly computable, as well as their multipliers) and it is well known that this is a dense subset of $J(P)$.

The following well known characterization of computable compact sets will be used in the sequel.

Proposition 2.1**.**

A compact set $K$ is computable if and only if it is simultaneously lower and upper computable.

As an immediate corollary, we obtain computability of some Julia sets (see [3]):

Corollary 2.2**.**

Let $P$ be a computable complex polynomial such that the filled Julia set $K(P)$ has empty interior. Then, the Julia set $J(P)=K(P)$ is computable.

However, in general, Julia sets need not be computable sets, as it was shown in [4]:

Theorem 2.3**.**

There exists computable parameters $\lambda$, with $|\lambda|=1$, such that the Julia set of the polynomial $\lambda z+z^{2}$ is not computable.

This result will play an essential role in the proof of our Main Theorem.

2.2. Some lemmas on computable maps

Here we gather a number of elementary results in computable analysis that will be required later in the paper.

Lemma 2.4**.**

Let $K\in\mathbb{C}$ be a compact set. Suppose $K$ is computable. Then there exists an algorithm which takes as input any finite list of rational balls $\{B_{n_{1}},\dots,\mathcal{B}_{n_{k}}\}$ and halts if and only if they cover $K$. In this case, we say that the relation $K\subset\bigcup_{i=1}^{k}B_{n_{i}}$ is semi-decidible.

The proof is straightforward and will be left to the reader.

Lemma 2.5** (Computable extension).**

Suppose $K\subset\mathbb{C}$ is a lower computable compact set. Let $\phi:K\to\mathbb{C}$ be a continuous function which is computable on a dense collection of points in $K$ which are uniformly computable. Suppose in addition that $\phi$ has a computable modulus of continuity, that is, there is a computable function $m:(0,a)\to(0,a)$ which is non deceasing and satisfies

[TABLE]

for all $x,y$ in $K$. Then, $\phi$ is computable.

Proof.

Let $\{z_{n}\}_{n\in\mathbb{N}}$ be the dense set on which $\phi$ is computable and let $z$ be any point in $K$. To compute $\phi(z)$ at a given precision, it suffices to compute $n$ such that $m(|z_{n}-z|)$ is small enough, and then output $\phi(z_{n})$ at a sufficiently high precision. ∎

Lemma 2.6** (Computable inverse).**

Let $\Omega\subset\mathbb{C}$ be a lower computable domain and let $K$ be a computable compact set in $\Omega$. Let $f:\Omega\to f(\Omega)$ be a homeomorphism which is computable on $K$. Then, the inverse $f^{-1}:f(K)\to K$ is also a computable homeomorphism.

Proof.

Let $y\in f(K)$ be a given point. We show how to compute $x$ from $y$ such that $f(x)=y$. The set $\Omega\setminus\{y\}$ is recursively enumerable, uniformly in $y$. Since $f$ is computable on $K$, there is a recursively open set $U_{y}$ such that $f^{-1}(\Omega\setminus\{y\})=U_{y}\cap K$. But since $f$ is a homeomorphism, $U_{y}\cap K=K\setminus\{x\}$ for some $x$. Note that now we can semi-decide whether $x\in B$ for any rational ball $B$, since this is the case if and only if $B$ together with $U_{y}$ form a covering of $K$. To compute $x$ at a given precision, just enumerate all balls with diameter less than this precision and semi-decide whether they contain $x$. ∎

Lemma 2.7** (Computable images).**

Let $\Omega\subset{\mathbb{C}}$ be a lower computable domain, and let $K\subset\Omega$ be a computable compact set. Let $f:\Omega\to f(\Omega)$ be a continuous function. Then, if $f$ is computable on $K$, its image $K^{\prime}=f(K)$ is a computable compact set.

Proof.

Since $K$ is in particular lower computable, we can uniformly compute a sequence of points $x_{n}\in K$ which is dense in $K$. The sequence $f(x_{n})\in K^{\prime}$ is therefore a computable sequence which is dense in $K^{\prime}$. This shows that $K^{\prime}$ is lower-computable. Since $K$ is upper-computable, its complement $\Omega\setminus K$ is a r.e. open set. We now show how one can enumerate a sequence of rational balls in ${\mathbb{C}}$ whose union exhausts ${\mathbb{C}}\setminus K^{\prime}$, thus proving computability of $K^{\prime}$. Let $B$ be a rational ball in ${\mathbb{C}}$ and denote by $\operatorname{cl}B$ its closure. It is easy to see that $\operatorname{cl}B$ is disjoint from $K^{\prime}$ if and only if $K^{\prime}\subset\mathbb{C}\setminus\operatorname{cl}B$. Note that $\mathbb{C}\setminus\operatorname{cl}B$ is a r.e. open set. Now, computability of $f$ on $K$ means that for any r.e. open $U^{\prime}\subset{\mathbb{C}}$ one can uniformly lower compute a set $U\subset{\mathbb{C}}$ satisfying $f^{-1}(U^{\prime}\cap K^{\prime})=U\cap K$. In particular, this implies that a r.e. open set $U^{\prime}\subset{\mathbb{C}}$ covers $K^{\prime}$ if and only if $U$ covers $K$, which is a semi-decidable relation when $K$ is computable. It follows that we can semi-decide if a given r.e. open set $U^{\prime}$ covers $K^{\prime}$. This implies that we can enumerate all the balls $B$ whose closure is disjoint from $K^{\prime}$, which constitutes a list of balls exhausting the complement of $K^{\prime}$, and the lemma is proved. ∎

3. Preliminaries on dynamics of complex polynomials.

In this section we introduce the tools of complex dynamics that will be used in the proof of our main result. We refer the reader to [13] for an in-depth introduction into the subject; the specific facts on the dynamics of $f_{b}$ can be found in [7].

3.1. Green’s function and Böttcher coordinate.

Let $d$ be a positive integer larger than 1, and let $f$ be a complex monic polynomial of degree $d$. Denote by by $K(f)$ the filled Julia set of $f$; that is, the set of all points $z$ in $\mathbb{C}$ whose forward orbit under $f$ is bounded in $\mathbb{C}$. The set $K(f)$ is compact and its complement is the connected set consisting of all points whose orbit converges to infinity in the Riemann sphere. Furthermore, we have $f^{-1}(K(f))=K(f)$ and $f(K(f))=K(f)$. The boundary $J(f)$ of $K(f)$ is the Julia set of $f$.

Recall that the Green’s function of $K(f)$ is the function $G_{f}:\mathbb{C}\to[0,+\infty)$ that is identically [math] on $K(f)$ and that for $z$ outside $K(f)$ is given by the limit

[TABLE]

The function $G_{f}$ is continuous, subharmonic, satisfies $G_{f}\circ f=d\cdot G_{f}$ on $\mathbb{C}$, and it is harmonic and strictly positive outside $K(f)$.

It is easy to see that the Julia set of a complex polynomial is connected if and only if every critical point has a bounded orbit. In this case, the unique conformal isomorphism

[TABLE]

conjugates $f$ to $z\mapsto z^{d}$. It is called the (normalized) Böttcher coordinate of $f$ at infinity and satisfies $G_{f}=\log|\varphi_{f}|$.

The definition of the Böttcher coordinate can be extended to the case of a disconnected Julia set as follows. It is well known that $K(f)$ is connected if and only if all critical values of $f$ lie inside $K(f)$. Let $\omega$ be the critical value of $f$ such that $G_{f}(\omega)$ is maximal. Then the domain

[TABLE]

is homeomorphic to a punctured disk. We then define $\varphi_{f}$ as the unique conformal isomorphism

[TABLE]

with $\varphi_{f}(\infty)=\infty$ and $\varphi_{f}^{\prime}(\infty)=1$. It is not hard to see that $\varphi_{f}$ still conjugates $f$ to $z\mapsto z^{d}$.

Let $S_{f}$ be the union of the critical points of $G_{f}$ in $\mathbb{C}\setminus K_{f}$ and the stable manifolds of the gradient flow of $G_{f}$ on $\mathbb{C}\setminus K_{f}$. Denote $V_{f}$ the open set $\mathbb{C}\setminus(K_{f}\cup S_{f})$. The Böttcher coordinate $\varphi_{f}$ extends to an analytic map $\varphi_{f}:V_{f}\to\mathbb{C}$ and satisfies $G_{f}(z)=\log|\varphi_{f}(z)|$ on $V_{f}$.

By definition, for $v>0$ the equipotential of level $v$ of $f$ is the set $G_{f}^{-1}(v)$. A Green’s line of $G_{f}$ is a smooth curve on the complement of $K(f)$ in $\mathbb{C}$ that is orthogonal to the equipotentials of $G_{f}$ and that is maximal with this property. Note that in the case when $K(f)$ is connected, every Green’s line must accumulate inside the Julia set $J(f)$. If $K(f)$ is not connected, some Green’s lines will terminate at escaping critical points of $f$ and their preimages.

Given $t$ in $\mathbb{R}/\mathbb{Z}$, the external ray of angle $t$ of $f$, denoted by $R_{f}(t)$, is the Green’s line of $G_{f}$ containing

[TABLE]

By the identity $G_{f}\circ f_{c}=d\cdot G_{c}$, for each $v>0$ and each $t$ in $\mathbb{R}/\mathbb{Z}$ the map $f$ maps the equipotential $v$ to the equipotential $d\cdot v$ and maps $R_{f}(t)$ to $R_{f}(d\cdot t)$. For $t$ in $\mathbb{R}/\mathbb{Z}$ the external ray $R_{f}(t)$ lands at a point $z$, if $G_{f}:R_{f}(t)\to(0,+\infty)$ is a bijection and if $G_{f}|_{R_{f}(t)}^{-1}(v)$ converges to $z$ as $v$ converges to [math] in $(0,+\infty)$. By the continuity of $G_{f}$, every landing point is in $J(f)=\partial K(f)$.

We use the following simple fact several times.

Lemma 3.1**.**

Let $f$ be a complex monic polinomial of degree $d\geq 2$, let $t$ be in $\mathbb{R}/\mathbb{Z}$ and suppose that the external ray $R_{f}(t)$ lands at a point $z_{0}$ of $K(f)$ which is not a critical value of $f$; so $f^{-1}(z_{0})$ consists of $d$ distinct points. Then each point of $f^{-1}(z_{0})$ is the landing point of precisely one of the external rays $R_{f}((t+k)/d)$, for $k\in\{1,2,\ldots,d-1\}$.

3.2. Dynamics of maps $z\mapsto\lambda z+bz^{2}+z^{3}$

Let us fix a $\lambda\in S^{1}$, and consider the family

[TABLE]

as above. For every $b$ in $\mathbb{C}$ the polynomial $f_{b}$ has two critical points (counted with multiplicity) and one indifferent fix point at 0. It is known that the presence of this indifferent fixed point forces at least one of these critical points to have a bounded orbit. When $b^{2}=3\lambda$, these two critical points are equal, and therefore both have a bounded orbit. It follows that $b\in\mathcal{M}_{\lambda}$. When $b^{2}\neq 3\lambda$, the two critical points are different.

In the case when $b\notin\mathcal{M}_{\lambda}$, let us denote $\omega_{1}$ the critical point with bounded orbit, and $\omega_{2}$ the other, escaping, critical point. The critical value $f_{b}(\omega_{2})$ has two preimages. We call co-critical point the preimage of $f_{b}(\omega_{2})$ which is different from $\omega_{2}$, and we denote it by $\omega^{\prime}_{2}$.

The map

[TABLE]

is a conformal isomorphism. For $v>0$ the equipotential $v$ of $\mathcal{M}_{\lambda}$ is by definition

[TABLE]

On the other hand, for $t$ in $\mathbb{R}/\mathbb{Z}$ the set

[TABLE]

is called the external ray of angle $t$ of $\mathcal{M}_{\lambda}$. We say that ${\cal R}_{\lambda}(t)$ lands at a point $b$ in $\mathbb{C}$ if $\Phi^{-1}(r\exp(2\pi it))$ converges to $z$ as $r\searrow 1$. When this happens $z$ belongs to $\partial\mathcal{M}_{\lambda}$.

Let $b_{1}$ be the parameter with potential $\eta=1/3$ and external angle $\theta=1/4$. Let $U$ denote the open set $\{z\in\mathbb{C}\mid G_{b_{1}}(z)<3G_{b_{1}}(\omega_{2})\}$. Note that the equipotential of level $3G_{b}(\omega_{2})$ is a real-analytic simple closed curve, and thus $U$ is a topological disk. The set $f_{b_{1}}^{-1}(U)$ is the set $\{z\in\mathbb{C}\mid G_{b_{1}}(z)<G_{b_{1}}(\omega_{2})\}$ which is bounded by a lemniscate pinching at the escaping critical point $\omega_{2}$. Let $U^{\prime}$ be the connected component of $f_{b_{1}}^{-1}(U)$ that contains the non-escaping critical point $\omega_{1}$. We will denote by $Q_{b_{1}}:U^{\prime}\to U$ the restriction of $f_{b_{1}}$ to $U^{\prime}$. The filled Julia set $K(Q_{b_{1}})$ is defined as the set of points in $U^{\prime}$ that remain in $U^{\prime}$ under iterations by $Q_{b_{1}}$. The Julia set $J(Q_{b_{1}})$ is the boundary of $K(Q_{b_{1}})$.

The following result, extracted from [7], states that $J(Q_{b_{1}})$ is a quasi-conformal copy of $J(\lambda z+z^{2})$.

Theorem 3.2**.**

There exist a quasi-conformal homeomorphism $\phi:\mathbb{C}\to\mathbb{C}$ which conjugates $Q_{b_{1}}$ to $\lambda z+z^{2}$ on their Julia sets.

It will not be necessary to give the definition of a quasi-conformal homeomorphism here since all what we will need is the following standard property of such maps (see e.g. [1]):

Proposition 3.3**.**

A quasi-conformal map $\phi$ from a topological disk $D$ into itself is Hölder-countinuous. More precisely, there exist constants $H,\alpha$ such that for every $x,y$ in $D$

[TABLE]

Buff and Henriksen also give a characterisation of $J(Q_{b_{1}})$ as the landing points of a particular set of dynamical rays that we now describe. Let $\Theta\subset{\mathbb{R}}/{\mathbb{Z}}$ be the set of angles $\theta$ such that for every integer $n\geq 0$ we have $3^{n}\theta\in[0,1/2]$ mod 1. It is a Cantor set forward invariant under multiplication by $3$. It is shown in [7] that for any $\theta\in\Theta$, the dynamical ray $R_{f_{b_{1}}}(\theta)$ does not bifurcate, and that the set defined by

[TABLE]

satisfies $\overline{X_{b_{1}}}\setminus X_{b_{1}}=J(Q_{b_{1}})\subset J(f_{b_{1}})$.

3.3. Julia sets in $\mathcal{M}_{\lambda}$

The parameter rays ${\cal R}_{\lambda}(1/6)$ and ${\cal R}_{\lambda}(1/3)$ both land at the parameter $b_{0}=4(\lambda-1)$, see [7]. The wake ${\cal W}_{0}$ is defined as the connected component of

[TABLE]

containing the ray ${\cal R}_{\lambda}(1/4)$.

Every dyadic number $\vartheta=(2p+1)/2^{k},k\geq 1$ and $0<2p+1<2^{k}$ can be expressed in a unique way as a finite sum

[TABLE]

where each $\varepsilon_{i},i=1,\ldots,k$ take the value 0 or 1. We define $\vartheta^{-}$ and $\vartheta^{+}$ by the formulae:

[TABLE]

Proposition 12 in [7].* Given any dyadic angle $\vartheta=(2p+1)/2^{k},k\geq 1,0<2p+1<2^{k},$ the two parameter rays ${\cal R}_{\lambda}(\vartheta^{-}/3)$ and ${\cal R}_{\lambda}(\vartheta^{+}/3)$ land at a common point $b_{\vartheta}$. Moreover,*

[TABLE]

The wake ${\cal W}_{\vartheta}$ is defined as the connected component of

[TABLE]

that contains the parameter ray ${\cal R}_{\lambda}(\theta)$ with $\theta$ in $]\vartheta^{-}/3,\vartheta^{+}/3[$. We now can define $\mathcal{X}_{\vartheta}$ to be the set of parameter rays

[TABLE]

and let $\mathcal{J}_{\vartheta}$ to be the set $\mathcal{J}_{\vartheta}=\overline{\mathcal{X}_{\vartheta}}\setminus\mathcal{X}_{\vartheta}$, where the closure is taken in ${\mathbb{C}}$.

Let $h:{\cal W}_{0}\times{\mathbb{C}}\to{\mathbb{C}}$ be a quasi-conformal extention of the holomorphic motion $h:{\cal W}_{0}\times X_{b_{1}}\to{\mathbb{C}}$ defined by $h_{b}(z)=\varphi_{b}^{-1}\circ\varphi_{b_{1}}(z)$. By [7, Lemma 13] the map $H_{\vartheta}:{\cal W}_{0}\to{\mathbb{C}}$ defined by

[TABLE]

is locally quasi-regular, and its restriction to the dyadic wake ${\cal W}_{\vartheta}$ is a locally quasi-conformal homeomorphism sending $\mathcal{J}_{\vartheta}$ to $J(Q_{b_{1}})$.

4. Proof of the Main Theorem

4.1. Computable Böttcher’s coordinate

Let

[TABLE]

be a polynomial of degree $d\geq 2$. Let $\varphi_{f}$ be the Böttcher’s coordinate of $f$ at infinity, and, as before, let

[TABLE]

where $S_{f}$ is the union of the critical points of $G_{f}$ in $\mathbb{C}\setminus K_{f}$ and the stable manifolds of the gradient flow of $G_{f}$ on $\mathbb{C}\setminus K_{f}$. The main result of this subsection is the following.

Proposition 4.1** (Computability of Böttcher’s coordinate).**

Let $f$ be a computable monic polynomial of degree $d\geq 2$. The open set $V_{f}$ is lower-computable and the Böttcher coordinate $\varphi_{f}$ is computable on $V_{f}$.

The proof of this proposition will be given after the following sequence of lemmas.

Lemma 4.2**.**

There is $R>0$ such that the Böttcher’s coordinate is computable on $\mathbb{C}\setminus\overline{\mathbb{D}}_{R}$.

Proof.

For $|z|$ sufficiently large the Böttcher coordinate can be written as a infinite product as follows:

[TABLE]

For example, if $R\geq\max\{4\sum_{j=1}^{d-1}|a_{j}|,4/3\}$, then by induction we have that for $|z|\geq R$,

[TABLE]

and thus, the principal value of the $d^{n+1}$-root is defined. Taking logarithm of the absolute value one can see that the corresponding series converges and thus, the product also converges. For computing the rate of convergence put

[TABLE]

Notice that

[TABLE]

This implies that

[TABLE]

and

[TABLE]

Thus,

[TABLE]

∎

Lemma 4.3**.**

The Green’s function $G_{f}$ is computable on $\mathbb{C}\setminus K_{f}$.

Proof.

For $z$ in $K_{f}$ and for every $k$ in $\mathbb{N}$ we have

[TABLE]

For $R\geq\max\{4\sum_{j=1}^{d-1}|a_{j}|,4/3\}$ we have by induction that if $|z|\geq R$ then

[TABLE]

This implies that

[TABLE]

∎

Proof of Proposition 4.1.

Let $R$ be a positive number such that the Böttcher coordinate is computable on $\mathbb{C}\setminus\overline{\mathbb{D}}_{R}$. Now consider the flow $(z,t)\to F(z,t)$ associated to the gradient vector field $\nabla G_{f}$ on the complement of $K_{f}$. Since $\nabla G_{f}$ is analytic the dependence on $z$ of the flow $F(z,t)$ is also analytic. Observe that for every $t>0$ we have that

[TABLE]

Thus, for every $z$ in $V_{f}$ there is $t\geq 0$ sufficiently large such that $|F(z,t)|>R$. It follows that the map

[TABLE]

is a holomorphic extension of the Böttcher coordinate to $V_{f}$ and so it must be equal to $\varphi_{f}$ on $V_{f}$. On the other hand, since $G_{f}$ is computable and analytic it follows that $\nabla G_{f}$ is also computable and effectively locally Lipschitz on the complement of $K_{f}$ (see [15, Theorem 2] and [8, Theorem 1]) which is recursively enumerable open. Thus, by [8, Theorem 3] for every $z\notin K_{f}$ the map $t\in[0,+\infty)\to F(z,t)$ is computable. This implies that we can semi-decide whether $|F(z,t)|>R$, which is equivalent to say that $V_{f}$ is lower-computable open. Moreover, using that $G_{f}$ is computable we conclude that the extension of the Böttcher coordinate on $V_{f}$ is also computable. ∎

4.2. Computable external rays and their landing points

Lemma 4.4** (Computable inverse branches).**

Let $f$ be a computable polynomial of degree $d$ and let $\beta$ be a fixed point of $f$ which is not a critical value. Then, one can uniformly compute positive real numbers $r_{0},r_{1},\dots,r_{d-1}$ and points $\beta=\beta_{0},\beta_{1},\dots,\beta_{d-1}$ in $\mathbb{C}$ such that:

• •

$f(\beta_{i})=\beta$* for $i=0,\dots,d-1$,*

• •

the open disks $D(\beta_{i},r_{i})$, $i=0,\dots,d-1$ are pairwise disjoint and

• •

$f$* restricted to each $D(\beta_{i},r_{i})$ is conformal and $f(D(\beta_{i},r_{i}))\subset D(\beta,r_{0})$ for $i=0,\dots,d-1$.*

Moreover, the inverse branches $g_{i}:f(D(\beta_{i},r_{i}))\subset D(\beta,r_{0})\to D(\beta_{i},r_{i})$ of $f$ are all computable, uniformly in $i$.

Proof.

Since $\beta$ is not a critical value, it has exactly $d$ different preimages, one of which is $\beta$ (since it is fixed). Let $\beta_{1},...,\beta_{d-1}$ be the other preimages. Since $f$ has finitely many critical values, all of which are computable, we can compute $r_{0}$ such that $D(\beta,r_{0})$ is at some positive distance away from the collection of critical values. Then, the open set $f^{-1}(D(\beta,r_{0}))$ consist of exactly $d$ connected components, each of which contains one of the $\beta_{i}$, $i=0,\dots,d-1$. Clearly, now we can compute numbers $r_{i}$ such that $D_{i}=D(\beta_{i},r_{i})$ is included in the component containing $\beta_{i}$. By construction, $f$ is conformal on each $D_{i}$. Moreover, by Theorem 4.5 from [9], $f|_{D_{i}}:D_{i}\to f(D_{i})\subset D(\beta,r_{0})$ and its inverse $g_{i}:f(D_{i})\to D_{i}$ is also computable, uniformly in $i$. The lemma is proved. ∎

Let $\theta\in\mathbb{R}/\mathbb{Z}$ be such that $R_{f}(\theta)$ lands. For an interval $I\subset(1,\infty)$ we will denote by $R_{f}^{\theta}(I)$ the ray segment defined by

[TABLE]

Lemma 4.5** (Effective landing).**

Let $f$ be a computable polynomial of degree $d$ and let $\beta$ be a fixed point of $f$ such that $|df(\beta)|>1$. Suppose that the dynamical ray $R_{f}(\theta)$ lands at $\beta$ and that $\theta$ is a computable angle. Then the set $R^{\theta}_{f}(1,2]\cup\{\beta(b)\}$ is a computable compact set.

Proof.

By Proposition 4.1 we can compute a dense sequence of points in $R_{f}^{\theta}(1,2]$, for instance by computing the sequence $\varphi^{-1}_{f}(q_{n})$ where $(q_{n})$ is a computable sequence of rationals which is dense in $(1,2]$. This sequence is of course also dense in $R^{\theta}_{f}(1,2]\cup\{\beta\}$, which is therefore a lower-computable set. We now show that it is also upper computable. Since $|df(\beta)|>1$, it follows that $\beta$ is an attracting fixed point for the the inverse branch $g=f^{-1}$ of $f$ that leaves $\beta$ fixed. Moreover, we can compute a neighbourhood $T_{\beta}$ of $\beta$ such that its closure $\overline{T_{\beta}}$ shrinks to $\{\beta\}$ under iterates by $g$. Indeed, we could take for instance $T_{\beta}$ to be the open disk centred at $\beta$ with radius $\ln(|df(\beta)|^{|df(\beta)|})$. Since $R_{f}^{\theta}(1,2]$ is lower-computable, we can find a point $z\in R_{f}^{\theta}(1,2]$ which belongs to $T_{\beta}$. Let $\eta(z)\in(1,2]$ be the level of the equipotential line containing $z$. That is, $\varphi_{f}^{-1}(\eta(z)e^{2\pi i\theta})=z$. Since the ray $R_{f}(\theta)$ is invariant under iterations by $g$, we have that $g^{i}(z)\in R_{f}^{\theta}(1,2]$, and since $\varphi_{f}$ conjugates $f$ to $z^{d}$, we obtain that

[TABLE]

Now, compute the ray segment $R_{f}^{\theta}[\eta(g(z)),\eta(z)]$ which goes from $g(z)$ to $z$ in $R_{f}^{\theta}(1,2]$. This is clearly a computable closed set. Thus, by lemma 2.4, we can semi-decide if it is contained in $T_{\beta}$. In a dovetail fashion, semi-decide whether $R_{f}^{\theta}[\eta(g^{i+1}z),\eta(g^{i}z)]$ is contained in $T_{\beta}$, for larger and larger $i$. Since $R_{f}(\theta)$ lands at $\beta$, this procedure must eventually stop for some $i^{*}$. Let $z^{*}=g^{i^{*}}(z)$ and denote by $I^{*}$ the ray segment $R_{f}^{\theta}[\eta(g(z^{*})),\eta(z^{*})]$. Let $d^{*}_{0}=\max\{d(z,\beta):z\in I^{*}\}$ so that $I^{*}$ is completely contained in $\overline{D}(\beta,d^{*}_{0})\subset T_{\beta}$. Since the region $T_{\beta}$ is trapping, it follows that all the iterates of $I^{*}$ by $g$ are contained in $\overline{D}(\beta,d^{*}_{0})$ and thus, so is $R_{f}^{\theta}(1,\eta(z^{*})]$. Let $d^{*}_{i}=\max\{d(z,\beta):z\in g^{i}(I^{*})\}$. Since $d^{*}_{i}\to 0$ as $i\to\infty$, one has that

[TABLE]

But the sets in the union of the right-hand side are all recursively enumerable open sets, uniformly in $i$. The lemma follows. ∎

Lemma 4.6** (Effective pairing).**

Let $f$, $\theta\in\mathbb{R}/\mathbb{Z}$ and $\beta$ be as in the previous lemma. Then, one can uniformly compute angles $\theta_{1},\dots,\theta_{d-1}$ in $\mathbb{R}/\mathbb{Z}$ and points $\beta_{1},\dots,\beta_{d-1}$ in $K(f)$ such that $\beta_{i}$ is precisely the landing point of $R_{f}(\theta_{i})$, for $i=1,\dots,d-1$.

Proof.

For $i=0,\dots,d-1$, let $\beta_{i}$, $r_{i}$ and $g_{i}$ as in Lemma 4.4. By Lemma 3.1, each $\beta_{i}$ is the landing point of exactly one of the rays $R_{f}(\frac{\theta+k}{d})$ for $k=1,\dots,d-1$. Since the angles $\frac{\theta+k}{d}$ are all computable, all we need to do is to decide, for each $\beta_{i}$, which of the rays is the one landing at $\beta_{i}$. By Lemma 4.5, the set $R_{f}^{\theta}(1,2]\cup\{\beta\}$ is a computable closed set. It is easy to see that one can compute $t$ such that $R=R_{f}^{\theta}(1,t]\cup\{\beta\}$ is contained in $D(\beta,r_{0})$. Now, for each $i$, the set $g_{i}(R)\subset D(\beta_{i},r_{i})$ is a computable closed set which is contained in the ray landing at $\beta_{i}$. To compute $\theta_{i}$, just choose any point $z\in g_{i}(R)$ different from $\beta_{i}$ and compute its external angle. This is $\theta_{i}$. ∎

4.3. Proof of the Main Theorem

The proof will follow from the following two lemmas.

Lemma 4.7**.**

Let $\phi$ be the quasi-conformal homeomorphism of Lemma 3.2, conjugating the $Q_{b_{1}}$ to $P_{\lambda}=\lambda z+z^{2}$ on their Julia sets. Suppose that $\lambda\neq 1$ is computable and $|\lambda|=1$. Then $\phi$ is computable on $J(Q_{b_{1}})$.

Proof.

Recall that by Lemma 3.2 there exist a quasi-conformal function $\phi:\mathbb{C}\to\mathbb{C}$ that conjugates $Q_{b}$ to $P(\lambda)$. We show how to compute this function on $J(Q_{b_{1}})$ by computing it on a dense set of points, and then invoking Lemma 2.5 with any bound on the uniform modulus of continuity of $\phi$, which exists because of the Holder property of quasi-conformal maps (see Proposition 3.3). The dense set will be given by the $\beta(b_{1})$ fixed point of $f_{b_{1}}$, together with all their preimages under $Q_{b_{1}}$. Since $\phi$ is a conjugacy, this set is sent to the set of pre-images of the $\beta(\lambda)$ fixed point of $P_{\lambda}$. Since these fixed points are computable, so are the sets of preimages. Thus, it is enough to show how to algorithmically decide, for a given preimage of $\beta(b)$, which preimage of $\beta(\lambda)$ it goes to. To achieve this, we use the external arguments of the points: on one side we start with $R_{0}$ (which lands at $\beta(b_{1})$), whose preimages are $R_{0}$ and $R_{1/3}$ (which lands at $\beta_{1}$, the preimage of $\beta$ different from it). Then $R_{1/9}$ (which lands at one preimage of $\beta_{1}$) and $R_{1/9+1/3}$ (which lands at the other preimage of $\beta_{1}$) and so on. On the other side these are $R_{0}$, then $R_{0}$ and $R_{1/2}$, then $R_{1/4}$ and $R_{1/4+1/2}$ and so on. By respecting the orientation, we can pair the angles on different sides. If, moreover, we were able to pair preimages of the $\beta$ fixed point with the external ray landing at them, we could then pair the preimages of the $\beta(b)$-fixed point with the corresponding preimages of the $\beta(\lambda)$ fixed point. But this is precisely given by Lemma 4.6, and so the proof is finished. ∎

Lemma 4.8**.**

The map $H_{\vartheta}:\mathcal{W}_{0}\to\mathbb{C}$ is computable on the closure of $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$. Moreover, the restriction of $H_{\vartheta}$ to the closure of $\mathcal{W}_{\vartheta}\setminus\mathcal{M}_{\lambda}$ has a computable inverse.

Proof.

By Proposition 4.1, the mappings $\varphi_{b}$, $\varphi_{b_{1}}$ and their inverses are computable on their domains $U_{b},U_{b_{1}}$, uniformly in $b$ and $b_{1}$. Hence, holomorphic motion $h=\varphi^{-1}_{b}\circ\varphi_{b_{1}}:U_{b_{1}}\to U_{b}$ is computable too. Recall that $H_{\vartheta}=h^{-1}(f^{k+1}(\omega_{2}(b)))$. Since $f^{k+1}(\omega_{2}(b))\in U_{b}$ for $b\notin\mathcal{M}_{\lambda}$, to prove computability of the map $H_{\vartheta}$ on $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$ it is enough to show that the critical point $\omega_{2}(b)$ is computable from $b$. But the collection of critical points is always computable from $b$, and we can identify the escaping one. Now, since $H_{\vartheta}$ is quasi-conformal, it has the holder property on some large enough ball and therefore, by Lemma 3.3, its computability can be extended up to the closure of $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$. It follows that the restriction of $H_{\vartheta}$ to the closure of $\mathcal{W}_{\vartheta}\setminus\mathcal{M}_{\lambda}$ is computable homeomorphism. Computable inverse will follow from Lemma 2.6. However, note that we can not apply it directly to the closure of $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$ because it may not be a computable set (it may not be upper-computable). Instead, we first note that since $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$ is a recursively enumerable open set, we can produce a sequence of computable compact sets whose union equals $\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda}$. We can then apply Lemma 2.6 to each of these set, which proves that the inverse is computable on $H_{\vartheta}(\mathcal{W}_{0}\setminus\mathcal{M}_{\lambda})$. But we can now apply Lemma 3.3 to this inverse, which proves that its computability can be extended to the closure, as was to be shown. ∎

Lemma 4.9**.**

If $\partial M_{\lambda}$ is upper-computable, then so is $\mathcal{J}_{\vartheta}$.

Proof.

Suppose $\partial M_{\lambda}$ is upper computable. We only need to show that we can enumerate a sequence of balls in $\mathbb{C}$ whose union exhaust the complement of $\mathcal{J}_{\vartheta}$. This complement is made by the complement of $\partial M_{\lambda}$, the complement of the mini-wake $\mathcal{W}_{\vartheta}$, and the collection of the unbounded components of $\mathbb{C}\setminus\overline{\mathcal{X}_{\vartheta}}$. The complement of $\partial M_{\lambda}$ is recursively enumerable by hypothesis.

Recall that the dynamical rays $R_{b_{1}}(0)$ and $R_{b_{1}}(1/2)$ both land at the $\beta$ fixed point of $f_{b_{1}}$. Thus, the curve $\{\beta\}\cup R_{b_{1}}(0)\cup R_{b_{1}}(1/2)$ cuts the plane into two connected components $V_{1}$ and $V_{2}$. Let $V_{2}$ be the one containing the escaping critical point $\omega_{2}$. To see that the complement of the mini-wake $\mathcal{W}_{\vartheta}$ is also recursively enumerable, we use the fact that $H_{\vartheta}$ maps $\mathcal{W}_{\vartheta}$ (respectively $\partial\mathcal{W}_{\vartheta}$) to $V_{1}$ (respectively $\partial V_{1}$). This is shown in the proof of Lemma 13 from [7]. In particular, $H_{\vartheta}$ maps the parameter rays $\mathcal{R}_{\lambda}(\vartheta^{-}),\mathcal{R}_{\lambda}(\vartheta^{+})$ to the dynamical rays $R_{b_{1}}(0)$ and $R_{b_{1}}(1/2)$. Now, let $B\subset\mathbb{C}$ be some computable ball containing $K(f_{b_{1}})$ and consider the set $B^{\prime}_{1}=\partial(B\cap V_{1})$. By Lemma 4.5, it is straightforward to see that the curve $B^{\prime}_{1}$ is a computable set. Since the inverse of $H_{\vartheta}$ is computable there (by Lemma 4.8), we see by Lemma 2.7 that $H^{-1}_{\vartheta}(B^{\prime}_{1})$ is also a computable set. It is now straightforward to see that the complement of the mini-wake $\mathcal{W}_{\vartheta}$ is recursively enumerable. It remains to show that the collection of unbounded components of $\mathbb{C}\setminus\overline{\mathcal{X}_{\vartheta}}$ is uniformly recursively enumerable. Recall that these components correspond to preimages by $H_{\vartheta}$ of the unbounded components of $\mathbb{C}\setminus\overline{X_{b_{1}}}$. But these components are precisely given by the preimages of $V_{2}$ by iterates of $f_{b_{1}}$. Note that, by Lemma 4.5 again, the set $B^{\prime}_{2}=\partial(B\cap V_{2})$ is computable, and using Lemma 4.4, we see that their preimages by $f_{b_{1}}$ are computable too and thus so are the preimages of these by $H_{\vartheta}$. Moreover, by taking any computable point in the bounded component of the complement of $B^{\prime}_{2}$, we see that the interior of this last collection can be uniformly enumerated, from which it is straightforward to see that the unbounded components of $\mathbb{C}\setminus\overline{\mathcal{X}_{\vartheta}}$ can be uniformly recursively enumerated, as it was to be shown.

∎

We are now ready to finish the proof of our main result.

Theorem 4.10**.**

There exists a computable $\lambda$ such that the bifurcation locus $M_{\lambda}$ is not computable.

Proof.

By Theorem 2.3, there exist a computable $\lambda$ such that $J(\lambda z+z^{2})$ is not computable. By Lemma 4.7 and Lemma 2.7, the Julia set $J_{b_{1}}$ of $Q_{b_{1}}$ is not computable either. We prove that the preimage of this set by $H_{\vartheta}$ is not computable. It is enough to show that the map $H_{\vartheta}$ is computable on this preimage. From the proof of Lemma 4.9 we see that $H_{\vartheta}$ is computable on the closure of $\mathbb{C}\setminus\mathcal{M}_{\lambda}$, which contains $\mathcal{J}_{\vartheta}$. By Lemma 2.7, $\mathcal{J}_{\vartheta}$ is not computable, and the Theorem now follows from Lemma 4.9. ∎