2.1 Real rays and the original formula

Consider the Riemann map $\Phi _M : \mathbb {C}~\setminus ~\overline {\mathbb {D}} \to \mathbb {C} \setminus \mathcal {M}$ , which is unique if we normalize it so that $\Phi _M(\infty ) = \infty $ , $\Phi _M'(\infty ) = 1$ . For each $\theta \in \mathbb {R}/\mathbb {Z}$ , we define the external ray of angle $\theta $ as

$$ \begin{align*}R_M(\theta) := \{ \Phi_M(r e^{2 \pi i \theta}) \ : \ r> 1 \}.\end{align*} $$

The ray $R_M(\theta )$ lands if there exists $\gamma (\theta ) := \lim _{r \to 1^+} \Phi _M(r e^{2 \pi i \theta })$ . It is conjectured that the ray $R_M(\theta )$ lands for any $\theta \in \mathbb {R}/\mathbb {Z}$ ; to circumvent this issue, we define as $\widehat {R}_M(\theta )$ the impression of $R_M(\theta )$ , and the set of real angles as

$$ \begin{align*}\widehat{\mathcal{R}} := \{ \theta \in \mathbb{R}/\mathbb{Z} \ : \ \widehat{R}_M(\theta) \cap \mathbb{R} \neq \emptyset \}.\end{align*} $$

Conjecturally, this is exactly the set of angles for which the corresponding ray lands on the real axis. For further details, see, for example, [Reference ZakeriZa].

The following lemma gives a characterization of $\widehat {\mathcal {R}}$ in terms of the dynamics of the doubling map.

Lemma 2.1. [Reference Douady, Branner and HjorthDo]

The set of real angles equals

$$ \begin{align*}\widehat{\mathcal{R}} := \{ \theta \in \mathbb{R}/\mathbb{Z} \ : \ |D^n(\theta) - 1/2| \geq |\theta - 1/2|\ \mathrm{for\ all}\ n \in \mathbb{N} \}.\end{align*} $$

Remark 2.2. By Lemma 2.1, one recognizes that a way to map any angle $\theta \in \mathbb {R}/\mathbb {Z}$ to the set of real angles is to consider the function

$$ \begin{align*}\Psi(x) := \tfrac{1}{2} + \inf_{k \geq 0} \big| D^k(x) - \tfrac{1}{2} \big|,\end{align*} $$

which indeed satisfies $\Psi (\mathbb {R}/\mathbb {Z}) \subseteq \widehat {\mathcal {R}}$ . As you can see from Figure 2, this function is discontinuous, while its graph ‘contains’ the graphs of all the magic formula functions $\varphi _H$ given by Theorem 2 (which are indeed continuous) for all components H.

Figure 2

The graph of the function $\Psi (x)$ discussed in Remark 2.2. The red round dots correspond to points on the graph of Douady’s magic formula function $T = \varphi _H$ for the main cardioid, the blue square dots to the image of $\varphi _H$ for the rabbit component, and the green triangular dots to the Kokopelli component.

2.2 Tuning and the Blé–Cabrera magic formula

Given any hyperbolic component H in the Mandelbrot set, let us recall that there is a tuning map which sends the main cardioid to the hyperbolic component H, and the Mandelbrot set to a small copy of itself which contains H.

In order to define the map precisely, let $\mathbf {a}_H = .\overline {a_1\cdots a_k} < \mathbf {b}_H =.\overline {b_1 \cdots b_k}$ denote the two external angles of rays landing at the root of H, and denote by $\mathbf {A}_H = a_1\cdots a_k$ and $\mathbf {B}_H = b_1 \cdots b_k$ the two corresponding finite binary words. Moreover, denote $\mathbf {a}^{\prime }_H = .\overline {a_1\cdots a_k b_1 \cdots b_k}$ and $\mathbf {b}^{\prime }_H = .\overline {b_1 \cdots b_k a_1\cdots a_k }$ . We call the set of external angles that belong to $(\mathbf {a}_H, \mathbf {a}^{\prime }_H]$ and land on H the upper part of the hyperbolic component, and the set of angles landing on H and belonging to $[\mathbf {b}^{\prime }_H, \mathbf {b}_H]$ the lower part.

The tuning map $\mathfrak {T}_H$ on the set of external angles is now defined as follows. If $\theta =. \epsilon _1 \epsilon _2 \cdots $ is the binary expansion of $\theta $ , then the angle $\mathfrak {T}_H(\theta )$ has binary expansion

$$ \begin{align*}\mathfrak{T}_H(\theta) = .A_{\epsilon_1} A_{\epsilon_2} \cdots\end{align*} $$

where $A_0 = \mathbf {A}_H$ , $A_1 = \mathbf {B}_H$ . Then the set $\Xi _H := \mathfrak {T}_H(\Xi )$ is the set of rays landing on the boundary of H.

If $\theta \in \mathbb {T} = \mathbb {R}/\mathbb {Z}$ has infinite binary expansion (that is, it is not a dyadic number) and $S = s_1\cdots s_n$ is a finite word in the alphabet $\{0,1\}$ , we denote by $S \cdot \theta $ the element of $\mathbb {T}$ given by

$$ \begin{align*}S \cdot \theta := \sum_{k=1}^n s_k 2^{-k} + 2^{-n} \theta,\end{align*} $$

that is, the point whose binary expansion is the concatenation of S and the binary expansion of $\theta $ . Recall that tuning behaves well with respect to concatenation; namely, for any $S, H$ and $\theta $ , we have

$$ \begin{align*}\mathfrak{T}_H(S \cdot \theta) = \mathfrak{T}_H(S) \cdot \mathfrak{T}_H(\theta).\end{align*} $$

We define the small real vein associated to H, and denote it by $\mathbb {R}_H$ , to be the real vein of the small Mandelbrot set associated to H. Let us denote by $\mathcal {R}_H$ the set of external angles of rays landing on $\mathbb {R}_H$ , and by $\widehat {\mathcal {R}}_H$ the set of external angles of rays whose impression intersects $\mathbb {R}_H$ . Clearly, $\mathbb {R}_{H_0} = \mathbb {R}$ if $H_0$ is the main cardioid, and $\mathcal {R}_{H_0} = \mathcal {R}$ .

Proposition 2.3. (Blé and Cabrera [Reference Blé and CabreraBC])

The map $T_H := \mathfrak {T}_H \circ T \circ \mathfrak {T}_H^{-1}$ maps $\Xi _H$ into the small real vein $\widehat {\mathcal {R}}_H$ . Moreover, this map (restricted to $\Xi _H$ ) is piecewise affine: in fact, it can be written in terms of binary expansions as

$$ \begin{align*}T_H(\theta) = \mathbf{B}_H \mathbf{A}_H \cdot \theta\end{align*} $$

if $\theta \in (\mathbf {a}_H, \mathbf {a}^{\prime }_H)$ , and

$$ \begin{align*}T_H(\theta) = \mathbf{A}_H \mathbf{B}_H \cdot \theta\end{align*} $$

if $\theta \in (\mathbf {b}^{\prime }_H, \mathbf {b}_H)$ .

Proof By construction, the tuning map $\mathfrak {T}_H : \Xi \setminus \{ 0\} \to \Xi _H \setminus \{ \mathbf {a}_H, \mathbf {b}_H \}$ is a bijection, so the first statement follows by looking at the following diagram:

Moreover, since $T(\theta ) = 10 \cdot \theta $ for $\theta <1/2$ and tuning behaves nicely with respect to concatenation, we have

$$ \begin{align*}T_H(\theta) = \mathfrak{T}_H ( 10 \cdot \mathfrak{T}_H^{-1}(\theta) ) = \mathfrak{T}_H(10) \cdot \theta,\end{align*} $$

which proves the second claim.

2.3 Veins, pseudocenters and complexity

Given a dyadic number $\theta \in S^1 = \mathbb {R}/\mathbb {Z}$ , we define its complexity as

$$ \begin{align*}\Vert \theta \Vert := \min \{ k \geq 0 \,{:}\, D^k(\theta) = 0 \kern-1pt\!\mod \kern-1pt1 \}.\end{align*} $$

Of course, if $\theta = {p}/{2^q}$ with p odd, then $\Vert \theta \Vert = q$ . Given an interval $(\theta ^-,\theta ^+)$ with $\theta ^- < \theta ^+$ , we define its pseudocenter $\theta _0$ to be the dyadic rational of lowest complexity inside the interval $(\theta ^-, \theta ^+)$ .

A pair of elements $(\theta ^-, \theta ^+)$ in $\mathbb {T}$ is a ray pair if the two external rays of angle $\theta ^-$ and $\theta ^+$ combinatorially land at the same parameter on the boundary of the Mandelbrot set. To be precise, one starts by defining a relation on $\mathbb {Q}/\mathbb {Z}$ by setting $\theta _1 \sim _{\mathbb {Q}} \theta _2$ if $R_M(\theta _1)$ and $R_M(\theta _2)$ land at the same point. Then one takes the transitive closure of this relation and finally its topological closure to define an equivalent relation $\sim $ on $\mathbb {R}/\mathbb {Z}$ . As constructed by Thurston [Reference Thurston, Schleicher and SelingerTh], there is a lamination $QML$ on the unit disk such that its induced equivalent relation is precisely $\sim $ . If the Mandelbrot set is locally connected, then $\theta _1 \sim \theta _2$ if and only if $R_M(\theta _1)$ and $R_M(\theta _2)$ land on the same point.

Ray pairs are partially ordered: in fact, we say that $(\theta _1^-, \theta _1^+) \prec (\theta _2^-, \theta _2^+)$ if the leaf $(\theta _1^-, \theta _1^+)$ separates $(\theta _2^-, \theta _2^+)$ from $0$ . Let us denote by $N(\theta )$ the number of ends of the Hubbard tree associated to the angle $\theta $ . Recall that if $\theta _1 \prec \theta _2$ , then $N(\theta _1) \leq N(\theta _2)$ .

A dyadic number $\theta _0$ defines a combinatorial vein in the Mandelbrot set as follows.

For instance, if $\theta _0 = \tfrac {1}{4}$ , then the vein extends all the way to $(\tfrac {1}{7}, \tfrac {2}{7})$ , since $N(\tfrac {1}{4}) = 3$ , and one can check easily that $N(\tfrac {1}{7}) = 3$ (the ‘rabbit’). On the other hand, if $\theta _0 = {3}/{16}$ , then the vein extends up to, but not including, the pair $({39}/{224}, {43}/{224})$ . Indeed, ${N({3}/{16}) = 5}$ , while $N({39}/{224}) = N({43}/{224}) = N({37}/{224}) = 3$ . Moreover, ${N(\theta _1) = N(\theta _2) = 5}$ for any ray pair $(\theta _1, \theta _2)$ which separates $({39}/{224}, {43}/{224})$ from ${3}/{16}$ , and ${N(\theta _1) = N(\theta _2) = 3}$ for any ray pair $(\theta _1, \theta _2)$ which separates $({39}/{224}, {43}/{224})$ from $0$ .

Note that condition (2) in the above definition is needed. For instance, if one considers the ray pair $(\theta _1, \theta _2) = ({77}/{255}, {78}/{255})$ , then its pseudocenter is $\theta _0 = 39/128$ . However, one checks that $N({77}/{255}) = 6 < 8 = N({39}/{128})$ (see Figure 3).

Figure 3

The Julia set for the ray pair $(\theta ^-, \theta ^+) = ({77}/{255}, {78}/{255})$ . One can see that the Hubbard tree for $\theta ^-$ and $\theta ^+$ has six ends, while the Hubbard tree for the pseudocenter $\theta _0 = {39}/{128}$ has eight ends. Here, $\Vert \theta _0 \Vert = 7$ , so the vein V associated to $\theta _0$ has $\delta _V = 6$ , but $\theta ^+$ does not belong to V. This shows that the formula of Proposition 2.6 cannot be extended beyond the vein V; indeed, the point in the forward orbit of $\theta ^+$ which is closest to $\tfrac {1}{2}$ is ${114}/{255} \in \widehat {\mathcal {R}}$ , while $D^{\delta _V}(\theta ^+) = {147}/{255}$ , so the formula would break down.

Definition 2.5. The complexity of a vein V with pseudocenter $\theta _0$ is given by

$$ \begin{align*}\delta_V = \Vert \theta_0 \Vert - 1.\end{align*} $$

The complexity $\delta _V$ equals:

• • the smallest k such that $D^k(\theta _0) = 1/2$ ;

• • the smallest k such that $D^k(\theta ^-)$ and $D^k(\theta ^+)$ lie on opposite connected components of the set $\mathbb {T}\setminus \{0, 1/2\}$ ;

• • the smallest k such that $f^k_c(c)$ lies on the spine $[-\beta , \beta ]$ of the Julia set of $f_c$ for c which belongs to the vein.

Clearly, $\delta _V = 0$ if and only if V is the real vein. Given a vein V in the upper half plane (that is with $\theta _0 <1/2$ ), its lower side is the set of angles $\theta ^+$ for which there exists $\theta ^- < \theta ^+$ such that $(\theta ^-, \theta ^+)$ belongs to the vein.

Proposition 2.6. Let V be a vein in the upper half plane, and $\delta _V$ be its complexity. Then for each ray pair $(\theta ^-, \theta ^+)$ which belongs to V we have

$$ \begin{align*}D^{\delta_V}(\theta^+) \subseteq \widehat{\mathcal{R}}.\end{align*} $$

In order to prove the proposition, we need the following lemma.

Lemma 2.7. Let $(\theta ^-, \theta ^+)$ be a ray pair, and $\theta _0$ be its pseudocenter, with $\theta _0 < 1/3$ (that is the hyperbolic component lies in the upper half plane, and not in the $1/2$ limb). Then we have

$$ \begin{align*}\theta^+ - \theta_0 < \theta_0 - \theta^-.\end{align*} $$

Proof Recall that, for each rational ${p}/{q}$ with $p, q$ coprime, there exists a unique set $C_{p/q}$ of q points on $\mathbb {T}$ such that the doubling map acts on $C_{p/q}$ with rotation number $p/q$ , that is, $D(x_i) = x_{i+p}$ , where the indices are considered modulo q. We know that the set $C_{p/q}$ is precisely the set of external rays which land on the $\alpha $ fixed point for $f_c$ , when c belongs to the $p/q$ -limb. Let us call sectors the q connected components of $\mathbb {T} \setminus C_{p/q}$ . In particular, we call critical sector $\mathcal {S}_1$ the smallest sector, and central sector $\mathcal {S}_0$ the largest sector. We say that an interval $I \subseteq \mathbb {T}$ is embedded if it is entirely contained within one sector. Let us consider the interval $I_0 = [\theta ^-, \theta ^+]$ , which is embedded by construction.

Let $k \geq 0$ be the smallest integer such that $I_{k+1} := D^{k+1}(I_0)$ is not embedded (it must exist, since D doubles lengths, and the length of an embedded interval is bounded above by a constant $c < 1$ ). Note that the doubling map is a homeomorphism between each non-central sector and its image, so $I_k$ must be contained in the central sector.

Let us now consider the set $\widehat {C}_{p/q} = C_{p/q} + 1/2$ , which is a subset of the central sector and equals the set of external rays which land on $-\alpha $ , the preimage of the $\alpha $ fixed point. Thus, let us call subsectors the connected components of $\mathcal {S}_0 \setminus \widehat {C}_{p/q}$ . In particular, exactly two subsectors $\mathcal {S}^-$ and $\mathcal {S}^+$ map to the critical sector $\mathcal {S}_1$ ; let us call them central subsectors. Since the rays $D^k(\theta ^-)$ and $D^k(\theta ^+)$ land together, then either they are both contained in the same subsector, or one is contained in one central subsector and the other one in the other. If they are contained in the same subsector, then the whole $I_k$ is contained in the same subsector, hence its image $I_{k+1}$ is still embedded, contradicting the definition of k. Thus, the two endpoints of $I_k$ are contained in two distinct central subsectors. Since $I_k$ is embedded, then it must contain $0$ ; moreover, since $0$ is the dyadic of lowest possible complexity, this means that $D^k(\theta _0) = 0$ . Now, by construction the images $D^{k+1}(\theta ^-)$ and $D^{k+1}(\theta ^+)$ lie in the critical sector $\mathcal {S}_1$ , which is contained inside the arc $[0, 1/2]$ since the hyperbolic component lies in the upper half plane. This means that

$$ \begin{align*}\ell([0, D^{k+1}(\theta^+)]) < 1/2 < \ell([D^{k+1}(\theta^-), 0])\end{align*} $$

(where $\ell $ denotes the length of the intervals) and, since $D^{k+1}$ is a homeomorphism when restricted to $[\theta _0, \theta ^+]$ and to $[\theta ^-, \theta _0]$ ,

$$ \begin{align*}2^{k+1} \ell([\theta_0, \theta^+]) = \ell(D^{k+1}[\theta_0, \theta^+]) = \ell([0, D^{k+1}(\theta^+)]) < 1/2,\end{align*} $$

and similarly,

$$ \begin{align*}2^{k+1} \ell([\theta^-, \theta_0]) = \ell(D^{k+1}[\theta^-, \theta_0]) = \ell([D^{k+1}(\theta^-), 0])> 1/2;\end{align*} $$

hence by comparing the previous two equations

$$ \begin{align*}\ell([\theta_0, \theta^+]) < \ell([\theta^-, \theta_0]),\end{align*} $$

which proves the claim.

2.4 Combinatorial Hubbard trees

Recall that every angle $\theta \in \mathbb {T}$ has an associated lamination on the disk which is invariant by the doubling map (see [Reference Thurston, Schleicher and SelingerTh]). The (two) longest leaves of the lamination are called major leaves, and their common image is called the minor leaf and will be denoted by m. Moreover, we let $\beta $ denote the leaf $\{0 \}$ , which we will take as the root of the lamination (the notation is due to the fact that the ray at angle $0$ lands at the $\beta $ -fixed point). The dynamics on the lamination is induced by the dynamics of the doubling map on the boundary circle. In particular, let us denote $f : \overline {\mathbb {D}} \to \overline {\mathbb {D}}$ to be a continuous function on the filled-in disk which extends the doubling map on the boundary $S^1$ . Let us denote by $\Delta $ the diameter of the circle which connects the boundary points at angles $\theta /2$ and $(\theta +1)/2$ . Then we shall also choose f so that it maps homeomorphically each connected component of $\mathbb {D}\setminus \Delta $ onto $\mathbb {D}$ .

Let $\mathcal {L}_1$ and $\mathcal {L}_2$ be two distinct leaves. Then we define the combinatorial segment $[\mathcal {L}_1, \mathcal {L}_2]$ as the set of leaves $\mathcal {L}$ of the lamination which separate $\mathcal {L}_1$ and $\mathcal {L}_2$ . Some simple properties of combinatorial segments are the following:

1. (a) if $\mathcal {L} \in [\mathcal {L}_1, \mathcal {L}_2]$ , then $[\mathcal {L}, \mathcal {L}_1] \subseteq [\mathcal {L}_1, \mathcal {L}_2]$ ;

2. (b) for any choice of leaves $\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3$ we have $[\mathcal {L}_1, \mathcal {L}_2] \subseteq [\mathcal {L}_3, \mathcal {L}_1] \cup [\mathcal {L}_3, \mathcal {L}_2]$ ;

3. (c) the image of $[\mathcal {L}_1, \mathcal {L}_2]$ equals

   $$ \begin{align*}f([\mathcal{L}_1, \mathcal{L}_2]) = \left\{ \begin{array}{@{}ll} \lbrack f(\mathcal{L}_1), f(\mathcal{L}_2) \rbrack & \mathrm{if}\ \Delta \mathrm{does\ not\ separate}\ \mathcal{L}_1\mathrm{and}\ \mathcal{L}_2, \\ \lbrack f(\mathcal{L}_1), m \rbrack \cup \lbrack f(\mathcal{L}_2), m \rbrack & \mathrm{if}\ \Delta \mathrm{separates}\ \mathcal{L}_1\mathrm{and}\ \mathcal{L}_2; \end{array} \right.\end{align*} $$

4. (d) in any case, for any leaves $\mathcal {L}_1, \mathcal {L}_1$ we have

   $$ \begin{align*}[f(\mathcal{L}_1), f(\mathcal{L}_2)] \subseteq f([\mathcal{L}_1, \mathcal{L}_2]) \subseteq \lbrack f(\mathcal{L}_1), m \rbrack \cup \lbrack f(\mathcal{L}_2), m \rbrack.\end{align*} $$

Finally, we shall say that a set S of leaves is combinatorially convex if whenever $\mathcal {L}_1$ and $\mathcal {L}_2$ belong to S, then the whole set $[\mathcal {L}_1, \mathcal {L}_2]$ is contained in S.

We now define

$$ \begin{align*}H_n := \bigcup_{0 \leq i \leq n} [\beta, f^i(m)]\end{align*} $$

and

$$ \begin{align*}H := \bigcup_{n \in \mathbb{N}} H_n.\end{align*} $$

We call the set H the combinatorial Hubbard tree of f, as it is a combinatorial version of the (extended) Hubbard tree.

Note that $N + 2$ coincides with the number of ends of the extended Hubbard tree.

Proof (1) Let us check that H is combinatorially convex. Suppose $\mathcal {L}_1, \mathcal {L}_2$ belong to H, so that, say, $\mathcal {L}_1 \in [\beta , f^i(m)]$ and $\mathcal {L}_2 \in [\beta , f^j(m)]$ . Then

$$ \begin{align*}[\mathcal{L}_1, \mathcal{L}_2] \subseteq [\mathcal{L}_1, \beta] \cup [\beta, \mathcal{L}_2] \subseteq [f^i(m), \beta] \cup [\beta, f^j(m)] \subseteq H\end{align*} $$

as required. In order to check that H is forward invariant, note that by point (d) and using that $f(\beta ) = \beta $ yields the inclusion

(1) $$ \begin{align} f([\beta, f^i(m)]) \subseteq [f(\beta), m] \cup [m, f^{i+1}(m)] \subseteq [\beta, m] \cup [\beta, f^{i+1}(m)] \end{align} $$

and the right-hand side is contained in H by construction, so $f(H) \subseteq H$ .

In order to check the minimality, let S be a combinatorially convex, forward-invariant set of leaves which contains $\beta $ and m. Then, by convexity, S contains $[\beta , m]$ . Moreover, by forward invariance for any $i \geq 0$ we have

$$ \begin{align*}S \supseteq f^i([\beta, m]) \supseteq [f^i(\beta), f^i(m)] = [\beta, f^i(m)],\end{align*} $$

thus $S \supseteq H$ .

(2) Note that by construction $H_N \subseteq H$ , and the same proof as in (1) shows that $H_N$ is combinatorially convex. In order to prove the claim it is thus enough to check that $H_N$ is forward invariant, because then by minimality we get $H_N \supseteq H$ and the claim is proven. To prove forward invariance, note that

$$ \begin{align*}f(H_N) = \bigcup_{0 \leq i \leq N} f([\beta, f^i(m)]) \subseteq H_N \cup [\beta, f^{N+1}(m)],\end{align*} $$

and since $f^{N+1}(m) \in H_N$ it follows that $[\beta , f^{N+1}(m)] \subseteq H_N$ , proving the claim.

Lemma 2.9. Let $m_1 < m_2$ , and $H_1, H_2$ be the corresponding combinatorial Hubbard trees. Then

$$ \begin{align*}H_1 \subseteq H_2.\end{align*} $$

As a corollary,

$$ \begin{align*}\#Ends(T_1) \leq \#Ends(T_2).\end{align*} $$

Lemma 2.10. Let $(\theta ^-, \theta ^+)$ be a ray pair, and $\theta _0$ its pseudocenter, with $\Vert \theta _0 \Vert = q$ . Then

$$ \begin{align*}\#Ends(T_{\theta^+}) = \#Ends(T_{\theta_0})\end{align*} $$

if and only if the arcs $I_k = (D^k(\theta ^-), D^k(\theta ^+))$ for $k = 0, \ldots , q - 1$ are disjoint.

3.1 Maximally diverse sequences

Definition 3.2. A sequence $(s_n) \in \{0, 1\}^{\mathbb {N}}$ is maximally diverse if the subsequences

$$ \begin{align*}(s_{i + n p})_{n \in \mathbb{N}}\end{align*} $$

with $p \geq 1$ , $0 \leq i \leq p-1$ , are all distinct.

Lemma 3.3. A sequence $(s_n)$ is maximally diverse if and only if the subsequences

$$ \begin{align*}(s_{i + np})_{n \in \mathbb{N}}\end{align*} $$

with $p \geq 1$ , $i \geq 0$ , are all distinct.

Corollary 3.4. If s is maximally diverse, then for any $i \geq 0$ and $p \geq 1$ the subsequence

$$ \begin{align*}(s_{i + np})_{n \in \mathbb{N}}\end{align*} $$

is maximally diverse.

Recall that an infinite sequence $(\epsilon _n)$ is Sturmian if there exists $\alpha \in (0, 1) \setminus \mathbb {Q}$ , $\beta \in \mathbb {R}$ such that

$$ \begin{align*}\epsilon_n = \lfloor (n+1) \alpha + \beta \rfloor - \lfloor n \alpha + \beta \rfloor - \lfloor \beta \rfloor\end{align*} $$

for all $n \geq 0$ .

3.2 Proof of Proposition 3.1

Let $\theta $ be an external angle of a ray landing on the boundary of the hyperbolic component H, of period p. If $\theta $ is irrational, then $\theta $ is the tuning of an irrational angle $\eta $ of a ray landing on the main cardioid. Thus, the binary expansion of $\theta $ is

$$ \begin{align*}\theta := S_{\epsilon_0} S_{\epsilon_1} S_{\epsilon_2} \ldots\end{align*} $$

where $(S_0, S_1)$ are the binary expansions of the two rays landing at the root of H, and $(\epsilon _n)$ is the binary expansion of the angle of a ray landing on the main cardioid. Thus, by [Reference Bullett and SentenacBS] the sequence $(\epsilon _n)$ is a Sturmian sequence, hence by Theorem 3.6 it is a maximally diverse sequence.

Hence, for any $\alpha = 0, \ldots , p-1$ we have that $(\theta _{\alpha + np})_{n \in \mathbb {N}}$ is either maximally diverse (if $(S_0)_\alpha \neq (S_1)_\alpha $ ) or constant (if $(S_0)_\alpha = (S_1)_\alpha $ ). Thus, if we consider the image $\sigma := \Phi _H(\theta )$ we have by Lemma 3.5 that $(\sigma _{\alpha + np})_{n \in \mathbb {N}}$ is either maximally diverse or eventually constant (e.c.).

Note that since H does not intersect the real axis, either $(S_0)_1 = (S_1)_1 = 0$ (if H lies in the upper half plane) or $(S_0)_1 = (S_1)_1 = 1$ (if H lies in the lower half plane). In both cases, the subsequence $(\theta _{1 + np})_{n \in \mathbb {N}}$ is e.c. Thus, there exists $j \geq 1$ such that the subsequence

$$ \begin{align*}(\sigma_{j + np})_{n \in \mathbb{N}}\end{align*} $$

is e.c.

Let us now suppose by contradiction that the angle $\sigma $ is renormalizable. This implies that there exist two words $Z_0, Z_1$ with $|Z_0| = |Z_1| = q$ such that $\sigma \in \{ Z_0, Z_1 \}^{\mathbb {N}}$ . Since by construction the angle $\sigma $ is real, the only possibility is that $(Z_1)_i = 1 - (Z_0)_i$ for any $i = 1, \ldots , q$ .

Case 1. Let us first assume that p is not a multiple of q. Then let $l := \text {lcm}(p, q)$ , and consider all remainder classes modulo l.

Now, let us pick $\alpha \in \{0, \ldots , p-1\}$ such that $(\sigma _{\alpha + np})$ is e.c. Find $\gamma $ in $\{0, \ldots , l-1\}$ such that $\gamma \equiv \alpha \!\mod\! p$ , and let $\beta \in \{0, \ldots , q-1\}$ such that $\beta \equiv \gamma \!\mod\! q$ . Then one can write $\gamma = \beta + m q$ with $0 \leq m \leq ({l}/{q}) - 1$ .

Suppose that one has

$$ \begin{align*}\sigma = Z_{\epsilon_0} Z_{\epsilon_1} \cdots\end{align*} $$

with $\epsilon _i \in \{0, 1\}$ .

If for some $\alpha \in \{ 0, \ldots , p-1 \}$ the sequence $(\sigma _{\alpha + np})_{n \in \mathbb {N}}$ is e.c., then so is its subsequence $(\sigma _{\gamma + n l})_{n \in \mathbb {N}}$ , which coincides with $((Z_{\epsilon _{m + nl/q}})_\beta )_{n \in \mathbb {N}}$ . Since $(Z_0)_\beta \neq (Z_1)_\beta $ , this implies $(\epsilon _{m + nl/q})_{n \in \mathbb {N}}$ is also e.c. On the other hand, if $(\sigma _{\alpha + np})_{n \in \mathbb {N}}$ is not e.c., then it is maximally diverse, hence by Lemma 3.4 so is $(\sigma _{\gamma + n l})_{n \in \mathbb {N}}$ , hence also $(\epsilon _{m + nl/q})_{n \in \mathbb {N}}$ is not e.c.

Let us now consider another $\alpha ' \in \{0, \ldots , p-1 \}$ which is q-equivalent to $\alpha $ . Then by the above discussion,

$$ \begin{align*}(\sigma_{\alpha + np})_{n \in \mathbb{N}}\, \mathrm{e.c.}\ \Leftrightarrow (\epsilon_{m + nl/q})_{n \in \mathbb{N}}\, \mathrm{e.c.}\ \Leftrightarrow (\sigma_{\alpha' + np})_{n \in \mathbb{N}} \,\mathrm{e.c.} \end{align*} $$

By Lemma 3.8, since we know that there exists at least some $\alpha $ for which $(\sigma _{\alpha + np})_{n \in \mathbb {N}}$ is e.c., then each sequence $(\sigma _{\alpha '+np})_{n \in \mathbb {N}}$ is e.c. for any $\alpha ' \in \{0, \ldots , p-1\}$ , hence $\sigma $ is also eventually periodic, thus the angle $\theta $ cannot be irrational.

Case 2. Finally, let us consider the case when p is a multiple of q. Then recall one can write

$$ \begin{align*}\sigma = P S_{\epsilon_0} S_{\epsilon_1} \cdots = Z_{\eta_0} Z_{\eta_1} \cdots\end{align*} $$

where P is a finite word of some length $k \geq 0$ .

Now let us suppose that k is not a multiple of q. Note that since H is contained in either the upper or the lower half plane, we have $(S_0)_1 = (S_1)_1$ . Moreover, we claim that $(S_0)_p \neq (S_1)_p$ : in fact, if one considers the external angles $\theta _0 = .\overline {S_0}$ and $\theta _1 = .\overline {S_1}$ , the number of periodic external rays of period dividing p and which lie in the interval $(\theta _0, \theta _1)$ is even, since on each landing point exactly two rays land. Hence, in their binary expansion the last digit of $S_0$ must be opposite to the last digit of $S_1$ .

Now, for each $i \geq 0$ , there is some index j such that $Z_{\eta _j}$ overlaps with both $S_{\epsilon _i}$ and $S_{\epsilon _{i+1}}$ . As the last part of $Z_i$ must coincide with the first part of $S_{\epsilon _{i+1}}$ , and since $S_0$ and $S_1$ start with the same symbol, this forces $Z_{\eta _j}$ to be either $Z_0$ or $Z_1$ , independently of i. However, since the last symbols of $S_0$ and $S_1$ are different, this means that $S_{\epsilon _i}$ is also fixed, hence the sequence $(\epsilon _i)$ must be e.c., which contradicts the irrationality of $\theta $ .

Finally, if k is multiple of q, then $S_0$ and $S_1$ are finite concatenations of $Z_0, Z_1$ , which means that $\theta $ lies already in the small copy of the Mandelbrot set with roots $\{Z_0, Z_1\}$ . Since $\{Z_0, Z_1\}$ represent a real pair, such a small copy of the Mandelbrot set lies in the $1/2$ -limb, which contradicts the fact that H is outside such limb and completes the proof of Proposition 3.1.

3.3 Proof of the magic formula