Relation to Complex Dynamics

Abstract

In this chapter we outline how rotation sets occur in the dynamical study of complex polynomial maps. Special attention is paid to the relation with the dynamics of complex quadratic and cubic polynomials. This link provides a geometric realization of rotation sets under m _d , whose abstract theory was developed in the previous chapters.

In this chapter we outline how rotation sets occur in the dynamical study of complex polynomial maps. Special attention is paid to the relation with the dynamics of complex quadratic and cubic polynomials. This link provides a geometric realization of rotation sets under m _d , whose abstract theory was developed in the previous chapters.

5.1 Polynomials and Dynamic Rays

We assume the reader is familiar with the basic notions of complex dynamics, as in [21]. Let $f:{\mathbb C} \to {\mathbb C}$ be a monic polynomial map of degree d ≥ 2. The filled Julia set K(f) is the union of all bounded orbits of f, and the Julia set J(f) is the topological boundary of K(f). Both are compact non-empty subsets of the plane. The complement ${\mathbb C} \smallsetminus K(f)$ is connected and can be described as the basin of infinity for f, that is, the set of all points whose orbits under f tend to ∞. The Green’s function of f is the continuous function $G: {\mathbb C} \to {\mathbb R}$ defined by

$$\displaystyle \begin{aligned} G(z)=\lim_{n \to \infty} \frac{1}{d^n} \log^+ |f^{\circ n}(z)|, \end{aligned}$$

which describes the escape rate of z to ∞ under the iterations of f. It is easy to see that G satisfies the relation

$$\displaystyle \begin{aligned} G(f(z))=d \, G(z) \end{aligned}$$

with G(z) = 0 if and only if z ∈ K(f). The Green’s function is harmonic in the basin of ∞, with critical points at all precritical points of f. In other words, ∇G(z) = 0 for some $z \in {\mathbb C} \smallsetminus K(f)$ if and only if f ^∘n(z) is a critical point of f for some n ≥ 0.

There is a unique conformal isomorphism β, defined in some neighborhood of ∞, which is tangent to the identity at ∞ (in the sense that lim_z→∞β(z)∕z = 1) and conjugates the action of f to that of the power map z↦z ^d:

$$\displaystyle \begin{aligned} \beta(f(z))=(\beta(z))^d \quad \text{for large} \quad |z|. \end{aligned}$$

We call β the Böttcher coordinate of f near ∞. The modulus of β is related to the Green’s function by the relation |β(z)| = e ^G(z) for large |z|. It is not hard to check that β is univalent in the domain $\{ z \in {\mathbb C}: G(z)>G_0 \}$, where

$$\displaystyle \begin{aligned} G_0 = \max \{ G(c): c \ \text{is a critical point of} \ f \}. \end{aligned}$$

In particular, if every critical point of f belongs to K(f), then G ₀ = 0 and β is a conformal isomorphism ${\mathbb C} \smallsetminus K(f) \to {\mathbb C} \smallsetminus \overline {{\mathbb D}}$. This happens precisely when K(f) is connected.

In what follows and unless otherwise stated we assume that K(f) is connected. In this case the inverse Böttcher coordinate $\psi =\beta ^{-1}: {\mathbb C} \smallsetminus \overline {{\mathbb D}} \to {\mathbb C} \smallsetminus K(f)$ is a conformal isomorphism which satisfies

$$\displaystyle \begin{aligned} \psi(z^d)=f(\psi(z)) \quad \text{for} \quad |z|>1. \end{aligned} $$

(5.1)

By the (dynamic) ray of f at angle $t \in {\mathbb T}$ we mean the real-analytic curve

$$\displaystyle \begin{aligned} R(t)= \psi \big( \{ r e^{2 \pi i t} : r>1 \} \big). \end{aligned}$$

The functional equation (5.1) shows that

$$\displaystyle \begin{aligned} f(R(t))=R(m_d(t)) \quad \text{for all} \quad t \in {\mathbb T}. \end{aligned} $$

(5.2)

We say that R(t) lands at z ∈ J(f) if lim_r→1ψ(re ^2πit) = z. It follows from (5.2) that if R(t) lands at z, then R(m _d (t)) lands at f(z). Similarly, if f has local degree k at w ∈ f ⁻¹(z), then there are k preimages {t ₁, …t _k } of t under m _d such that each R(t _i ) lands at w. A ray may or may not land, but the set of angles t for which R(t) lands has full Lebesgue measure on the circle.

The impression $\hat {R}(t)$ of the ray R(t) is the set of all $w \in {\mathbb C}$ for which there is a sequence $z_n \in {\mathbb C} \smallsetminus \overline {{\mathbb D}}$ such that z _n  → e ^2πit and ψ(z _n ) → w. It is not hard to check that $\hat {R}(t)$ is a non-empty compact connected subset of J(f). Every point of the Julia set belongs to at least one impression. We say that the impression $\hat {R}(t)$ is trivial if it reduces to a single point {z}. In this case, R(t) necessarily lands at z (a landing ray, however, may well have a non-trivial impression). Furthermore, it is easily seen that

$$\displaystyle \begin{aligned} \limsup_{n \to \infty} \hat{R}(t_n) \subset \hat{R}(t) \quad \text{whenever} \quad t_n \to t. \end{aligned} $$

(5.3)

(As usual, the limsup on the left is the set of all $p \in {\mathbb C}$ such that every neighborhood of p meets infinitely many of the $\hat {R}(t_n)$.) We will also use the following separation property later on: Suppose the rays R(t′), R(t″) land at z and W is one of the two connected components of ${\mathbb C} \smallsetminus (R(t') \cup R(t'') \cup \{ z \})$. If a third ray R(t) is contained in W, then $\hat {R}(t) \subset W \cup \{z \}$.

A point z ∈ K(f) is the landing point of two or more rays if and only if $K(f) \smallsetminus \{ z \}$ is disconnected. More precisely, z has 2 ≤ n ≤∞ distinct rays landing on it if and only if $K(f) \smallsetminus \{ z \}$ has n connected components [18]. If z has finite forward orbit under f, the number of rays landing on it can be arbitrarily large (see the case of a parabolic fixed point below). But if the forward orbit of z is infinite, there is an upper bound C(d) for the number of rays that can land at z (one can take C(d) = 2^d, and the bound improves to C(d) = d if z is not precritical [15]).

The multiplier of a fixed point ζ = f(ζ) is the derivative f′(ζ). We call ζ attracting, repelling, or indifferent, according as the modulus |f′(ζ)| is less than, greater than, or equal to 1. An indifferent fixed point is called parabolic if its multiplier is a root of unity. The multiplier and type of a periodic point ζ of period n can be defined analogously by treating ζ as a fixed point of the iterate f ^∘n.

Suppose the angle $t \in {\mathbb T}$ is periodic of period q ≥ 1 under m _d , so t is rational of the form i∕(d ^q − 1). According to the Douady-Hubbard landing theorem [21], the ray R(t) lands at a periodic point of f with period dividing q, and this periodic point is necessarily repelling or parabolic. Conversely, every repelling or parabolic periodic point of f is the landing point of finitely many rays whose angles are periodic under m _d of the same period.

As a special case, if $u_i=i/(d-1) \ (\operatorname {mod} \ {\mathbb Z})$, it follows that for each 0 ≤ i ≤ d − 2 the fixed ray R(u _i ) lands at a repelling or parabolic fixed point ζ _i  = f(ζ _i ). When ζ _i is parabolic, the multiplier f′(ζ _i ) is necessarily 1. Of course the fixed points ζ ₀, …, ζ _d−2 need not be distinct.

The study of dynamic rays when K(f) is disconnected is a bit more complicated (an example of this case will be briefly discussed in Sect. 5.4). In this case at least one critical point of f escapes to ∞ and the Green’s function G has infinitely many critical points outside K(f). We can still define the dynamic rays $\{ R(t) \}_{t \in {\mathbb T}}$ partially near ∞ by pulling back the radial lines under the Böttcher coordinate

$$\displaystyle \begin{aligned} \beta: \{ z \in {\mathbb C}: G(z)>G_0 \} \to \{ z: |z|>e^{G_0} \}. \end{aligned}$$

These partial rays are the trajectories of the gradient vector field ∇G near ∞, so they can be extended in backward time. Such an extended trajectory either avoids the critical points of G and tends to K(f), or it eventually tends to such a critical point (namely an escaping precritical point of f). We call the ray smooth or bifurcated accordingly. For all but countably many $t \in {\mathbb T}$ the ray R(t) is smooth. In this case R(m _d (t)) is also smooth and the relation (5.2) holds. On the other hand, for a countably infinite set of angles t the ray R(t) is bifurcated. Under the iterations of f every bifurcated ray eventually maps to a smooth ray passing through a critical value of f.

5.2 Rotation Sets and Indifferent Fixed Points

This section will study polynomial maps of degree d ≥ 2 with connected Julia set which have an indifferent fixed point of multiplier e ^2πiθ ≠ 1. Every such map is affinely conjugate to a monic polynomial of the form

$$\displaystyle \begin{aligned} f : z \mapsto e^{2\pi i \theta} \, z + a_2 \, z^2 + \cdots + a_{d-1} \, z^{d-1} + z^d, \end{aligned} $$

(5.4)

where the indifferent fixed point is placed at the origin. We consider two cases depending on the nature of the fixed point 0.

The parabolic case. First suppose 0 is a parabolic fixed point so θ is rational of the form p∕q in lowest terms. Then there are finitely many rays landing at 0, each being periodic of period q. We can label these rays as

$$\displaystyle \begin{aligned} R(t_1), R(t_2), \ldots, R(t_{Nq}) \end{aligned}$$

where N ≥ 1 and 0, t ₁, …, t _Nq are in positive cyclic order. Using the form of the multiplier, it is easily seen that f(R(t _j )) = R(t _j+Np), or m _d (t _j ) = t _j+Np for every j, where as usual the indices are taken modulo Nq. It follows that {t ₁, …, t _Nq } is the union of N disjoint q-cycles under m _d , each with the combinatorial rotation number p∕q.

The following lemma ties up the situation with rotation sets:

Lemma 5.1

The set X of the angles $t \in {\mathbb T}$ for which the ray R(t) lands at 0 is a rotation set under m _d with ρ(X) = p∕q.

Proof

Label X = {t ₁, …, t _Nq } as above. For 1 ≤ i ≤ N, let C _i denote the q-cycle

$$\displaystyle \begin{aligned} t_i \mapsto t_{i+Np} \mapsto t_{i+2Np} \mapsto \ldots \mapsto t_{i+(q-1)Np} \end{aligned}$$

under m _d . Evidently X is the disjoint union of C ₁, …, C _N and these cycles are superlinked in the sense of Sect. 2.3. By Lemma 2.25, X is a rotation set with ρ(X) = ρ(C _i ) = p∕q. □

The deployment invariant of X can be described dynamically as follows. Two adjacent rays R(t _j ) and R(t _j+1) together with their common landing point 0 divide the plane into two open sectors. By definition, the (dynamic) wake W _j is the sector that contains the rays R(t) with t ∈ (t _j , t _j+1) (thus, W _j is the sector defined by going counter-clockwise from R(t _j ) to R(t _j+1)). The gap I _j  = (t _j , t _j+1) of X corresponds to the part of the boundary of the wake W _j on the circle at ∞. By Lemma 2.13, the multiplicity n _j of I _j is the number of fixed rays that are contained in W _j . It is also the number of the critical points of f in W _j (see [12], where this invariant is called the “critical weight” of W _j , and compare Theorem 5.10 for a similar case). In particular, I _j is a major gap if and only if W _j contains a fixed point ζ _i , or equivalently a critical point. As there are d − 1 fixed rays, there are at most d − 1 indices 1 ≤ j ≤ Nq for which n _j  ≠ 0. Form the non-decreasing list of integers 0 ≤ s ₁ ≤ s ₂ ≤⋯ ≤ s _d−1 = Nq in which each index 1 ≤ j ≤ Nq appears n _j times. It then follows from Lemma 3.5 that (s ₁, …, s _d−1) is the signature s(X) as defined in Sect. 3.2 and therefore (s ₁∕(Nq), …, s _d−1∕(Nq)) is the cumulative deployment vector σ(X).

Since the multiplier of the fixed point 0 is a q-th root of unity, the q-th iterate of f has the local expansion

$$\displaystyle \begin{aligned} f^{\circ q}(z) = z + a \, z^m + O(z^{m+1}) \quad \text{for some} \quad a \neq 0 \quad \text{and} \quad m>1. \end{aligned}$$

The integer m, the algebraic multiplicity of 0 as the root of the equation f ^∘q(z) − z = 0, is necessarily of the form kq + 1 for some 1 ≤ k ≤ N. According to Leau and Fatou [21], there are bounded Fatou components U ₁, …, U _kq arranged as kq “petals” around the common boundary point 0. If we choose labeling counter-clockwise, we have f(U _j ) = U _j+kp for every j, taking indices modulo kq, so the U _j are permuted with combinatorial rotation number p∕q. Every point in the union U ₁ ∪⋯ ∪ U _kq has an infinite orbit that tends to 0. Conversely, every infinite orbit converging to 0 must eventually enter this union. It follows from this local picture that the petal number kq of the parabolic fixed point is bounded above by the ray number Nq. The bound N ≤ d − 1 of Theorem 2.27 now shows that

$$\displaystyle \begin{aligned} q \leq \, \text{petal number} \ kq \leq \, \text{ray number} \ Nq \leq (d-1)q. \end{aligned}$$

In the quadratic case d = 2 it follows that the petal number and ray number are both q, while in the cubic case d = 3 these numbers can be q or 2q (see Fig. 5.1 for the case (k, N) = (1, 1) and (1, 2), and Fig. 5.9 for the case (k, N) = (2, 2)).

Fig. 5.1

Examples of parabolic points with multiplier λ = e ^2πi∕3 and petal number 3. Left: The cubic z↦λz − (0.04 + 0.85i)z ² + z ³ with ray number 3. Right: The cubic z↦λz + (0.23 − 0.20i)z ² + z ³ with ray number 6. The critical points c, c′ are marked as white dots

The “good” Siegel case. Now suppose 0 is a linearizable fixed point, so it belongs to a bounded Fatou component Δ in which the action of f is conjugate to the irrational rotation z↦e ^2πiθz. The domain Δ is called the Siegel disk of f centered at 0. We will assume that the boundary ∂Δ is a Jordan curve containing at least one critical point of f. This is certainly the case if θ is an irrational number of bounded type, that is, if the partial quotients in the continued fraction expansion θ = [a ₁, a ₂, a ₃, …] form a bounded sequence (compare [8] and [31]).¹ To avoid topological complications and focus on the combinatorial aspects of the constructions, we further make the following assumption:

The Limb Decomposition Hypothesis

There is a countable collection of disjoint non-trivial compact connected subsets of K(f), called limbs , such that

(LD1):

K(f) is $\overline {\varDelta }$ union all the limbs,

(LD2):

Each limb meets $\overline {\varDelta }$ at a single point on ∂Δ called its root ,

(LD3):

For each ε > 0 there are at most finitely many limbs with diameter > ε.²

We denote by L(p) the limb with root p ∈ ∂Δ.

Lemma 5.2

A point p ∈ ∂Δ is a root if and only if $K(f) \smallsetminus \{ p \}$ is disconnected.

Proof

For every root p the non-empty set $L(p) \smallsetminus \{ p \}$, which is clearly closed in $K(f) \smallsetminus \{ p \}$, is also open in there by the condition (LD3) above. It follows that $K(f) \smallsetminus \{ p \}$ is disconnected. Conversely, if $K \smallsetminus \{ p \}$ is disconnected for some p ∈ ∂Δ, there are two distinct rays landing at p. These rays together with their landing point divide the plane into two open sectors, one containing Δ and the other containing a non-trivial subset of K(f) which necessarily lies in a single limb. It easily follows that p is the root of this limb. □

Lemma 5.3

The set of roots is backward-invariant and therefore everywhere dense on ∂Δ.

Proof

Take a root p and let z be the unique point on ∂Δ such that f(z) = p. There are small neighborhoods U of z and U′ of p such that f : U → U′ acts as the power w↦w ^k for some k ≥ 1. Take two distinct rays landing at p, take their intersections with U′ and pull them back under f to obtain 2k ≥ 2 arcs in U landing at z. Each such arc is necessarily contained in a ray because of the functional equation (5.1). It follows that $K \smallsetminus \{ z \}$ is disconnected and therefore z is a root by Lemma 5.2. This proves backward-invariance of roots. Density of roots is now immediate since f| _∂Δ  : ∂Δ → ∂Δ is conjugate to an irrational rotation. □

Every root p has infinite forward orbit since f| _∂Δ is conjugate to an irrational rotation. It follows that there are at least 2 and at most 2^d rays landing at p. These rays together with their landing point p divide the plane into finitely many open sectors. There is a unique sector that contains Δ which we call the co-wake with root p and denote by V (p). The complement $W(p)={\mathbb C} \smallsetminus \overline {V(p)}$ is called the (dynamic) wake with root p. Thus W(p) is bounded by two rays landing at p and contains $L(p) \smallsetminus \{ p \}$ (see Fig. 5.2). Notice that distinct wakes are disjoint. Every point in the plane is either in $\overline {\varDelta }$, or in a unique wake, or else on a unique ray which is outside all wakes.

Fig. 5.2

The wake W(p) with the root p on the boundary of the Siegel disk Δ

Lemma 5.4

Every ray R(t) that is outside all wakes lands at a point z ∈ ∂Δ. Moreover,

• (i) If z is not a root, then $\hat {R}(t) = \{ z \}$.

• (ii) If z is a root, then $\hat {R}(t) \subset L(z)$ so $\hat {R}(t) \cap \partial \varDelta = \{ z \}$.

Proof

Let us first make the extra assumption that the ray R = R(t) is not a boundary ray of any wake. Suppose the impression $\hat {R}$ contains a point z∉∂Δ. Then z belongs to a limb L(p), and since z ≠ p, we have z ∈ W(p). Since by our assumption R is disjoint from $\overline {W(p)}$, it must be contained in the co-wake V (p). But then $\hat {R} \subset V(p) \cup \{ p \}$, which implies z ∈ V (p), contradicting z ∈ W(p). This proves $\hat {R} \subset \partial \varDelta$. If the impression $\hat {R}$ is non-trivial, by connectivity it must contain an open subarc T ⊂ ∂Δ. By Lemma 5.3, there are distinct roots p, p′∈ T. The open set ${\mathbb C} \smallsetminus (\overline {W(p)} \cup \overline {W(p')} \cup \overline {\varDelta })$ has two connected components and R is contained in one of them, say H. It follows that $T \subset \hat {R} \subset \overline {H} \cap \partial \varDelta$. But the intersection $\overline {H} \cap \partial \varDelta$ is one of the two closed subarcs of ∂Δ with endpoints p, p′, neither of which contains the open arc T. The contradiction proves that $\hat {R}$ is a single point on ∂Δ.

Now consider the case where R is one of the two boundary rays of a wake W(z). An argument similar to the above paragraph shows that $\hat {R} \subset L(z) \cup \partial \varDelta$. If $\hat {R}$ contained a point of ∂Δ other than z, it would have to contain a non-degenerate open arc in ∂Δ. A similar argument as before would then yield a contradiction. This shows $\hat {R} \subset L(z)$ and completes the proof. □

Corollary 5.5

Every non-root z ∈ ∂Δ belongs to the impression of a unique ray. This ray has trivial impression and therefore lands at z.

Proof

Let R(t) be any ray whose impression contains z. Then R(t) is outside all wakes since R(t) ⊂ W(p) would imply $\hat {R}(t) \subset W(p) \cup \{ p \}$ which in turn would imply z = p is a root. It follows from the previous lemma that $\hat {R}(t) = \{ z \}$. To see uniqueness, simply note that if $\hat {R}(s)$ also contained z for some s ≠ t, then by the above observation $\hat {R}(s) = \{ z \}$. As the landing point of two distinct rays, z would disconnect K(f) and therefore would be a root by Lemma 5.2. □

Let $\iota : {\mathbb C} \to \overline {\varDelta }$ be the map that is the identity on $\overline {\varDelta }$, sends every wake to its root and sends every ray outside all wakes to its landing point (Lemma 5.4).

Lemma 5.6

$\iota : {\mathbb C} \to \overline {\varDelta }$ is a retraction.

Proof

We need only check continuity of ι at every point z that does not belong to Δ or any wake. First consider the easier case where z ∈ ∂Δ. Take a sequence $z_n \notin \overline {\varDelta }$ that tends to z. Each z _n belongs to a limb L(p _n ) and we may assume that these limbs are distinct. Since $\operatorname {diam}(L(p_n)) \to 0$ by (LD3), it easily follows that ι(z _n ) = p _n  → z = ι(z).

Now consider the case where z belongs to a ray R(t) outside all wakes. Take any sequence z _n  → z. For large n, each z _n belongs to a unique ray R(t _n ), where t _n  → t. We distinguish two cases: Case 1 After passing to a subsequence, every ray R(t _n ) is outside all wakes. Then, by (5.3) and Lemma 5.4,

$$\displaystyle \begin{aligned} \limsup_{n \to \infty} \{ \iota(z_n) \} = \limsup_{n \to \infty} \hat{R}(t_n) \cap \partial \varDelta \subset \hat{R}(t) \cap \partial \varDelta = \{ \iota(z) \}.\end{aligned} $$

This proves ι(z _n ) → ι(z).

Case 2 After passing to a subsequence, each R(t _n ) lies in some wake W(p _n ). Then the impression $\hat {R}(t_n)$ is contained in the limb L(p _n ) whose diameter tends to 0 as n →∞. Hence $\limsup _{n \to \infty } \hat {R}(t_n)$ coincides with the set of all accumulation points of the sequence of roots {p _n  = ι(z _n )}. Again, by (5.3) and Lemma 5.4, $\limsup _{n \to \infty } \hat {R}(t_n) \subset \hat {R}(t) = \{ \iota (z) \}$, and we conclude that ι(z _n ) → ι(z). □

Recall that for 0 ≤ i ≤ d − 2 the fixed point ζ _i  ∈ J(f) is the landing point of the fixed ray R(u _i ). Let w _i  = ι(ζ _i ) ∈ ∂Δ. Since the ζ _i do not belong to $\overline {\varDelta }$, they lie in wakes, so every w _i must be a root. We call {w ₀, …, w _d−2} the marked roots of f. Take the unique conformal isomorphism $h: \varDelta \to {\mathbb D}$ which fixes 0 and sends w ₀ to 1. According to Carathéodory, since ∂Δ is a Jordan curve, h extends to a homeomorphism between the closures [21]. Note that $h \circ f \circ h^{-1}: {\mathbb D} \to {\mathbb D}$ fixes 0 and has derivative e ^2πiθ at the origin, so by the Schwarz lemma,

$$\displaystyle \begin{aligned} h(f(z))=e^{2 \pi i \theta} h(z) \quad \text{for all} \quad z \in \varDelta. \end{aligned}$$

We define the internal angle of a point z ∈ ∂Δ as the unique $\alpha \in {\mathbb T}$ such that h(z) = e ^2πiα. By the above conjugacy relation, the internal angle of f(z) will then be $\alpha +\theta \ (\operatorname {mod} \ {\mathbb Z})$.

Let α ₁, α ₂, …, α _d−1 denote the internal angles of the marked roots w ₁, w ₂, …, w _d−1 = w ₀. The following is the analog of Lemma 5.1:

Theorem 5.7

The set X′ of all angles $t \in {\mathbb T}$ for which the ray R(t) lands on ∂Δ contains a unique minimal rotation set X for m _d , with ρ(X) = θ. Moreover, the cumulative deployment vector of X satisfies

$$\displaystyle \begin{aligned} \sigma(X)=(\alpha_1,\ldots, \alpha_{d-1}) \qquad (\operatorname{mod} \ {\mathbb Z}^{d-1}). \end{aligned} $$

(5.5)

The proof will show that the difference $X' \smallsetminus X$ consists of at most countably many isolated points.

Proof

For each root p ∈ ∂Δ let I(p) be the open interval of angles $t \in {\mathbb T}$ for which R(t) ⊂ W(p). Set $X = {\mathbb T} \smallsetminus \bigcup _p I(p)$. By Lemma 5.4 the compact set X is contained in X′ and the difference $X' \smallsetminus X$ consists of the at most countable set of angles of rays within some wake that land at a root.

Let $\psi : {\mathbb C} \smallsetminus \overline {{\mathbb D}} \to {\mathbb C} \smallsetminus K(f)$ be the inverse Böttcher coordinate of f near ∞. Define $\varphi : {\mathbb T} \to {\mathbb T}$ by letting φ(t) be the internal angle of the point ι(ψ(2e ^2πit)) ∈ ∂Δ. The map φ is continuous by the previous lemma, and is surjective by Corollary 5.5. Using the fact that distinct rays cannot cross, it is not hard to see that φ is monotone of degree 1, with the collection of intervals {I(p) : p is a root} as its plateaus. If R(t) lands at z ∈ ∂Δ with internal angle α, then R(m _d (t)) lands at f(z) with internal angle α + θ. This proves

$$\displaystyle \begin{aligned} \varphi \circ m_d = r_{\theta} \circ \varphi \quad \text{on} \quad X. \end{aligned}$$

Furthermore, if the fiber φ ⁻¹(α) is non-trivial, then h ⁻¹(e ^2πiα) is a root, so its preimage h ⁻¹(e ^{2πi(α−θ)}) is also a root by Lemma 5.3, which proves the fiber φ ⁻¹(α − θ) is non-trivial as well. It now follows from Theorem 2.35 that X is a minimal rotation set for m _d with ρ(X) = θ, and φ is the canonical semiconjugacy associated with X.

The claim (5.5) on σ(X) follows from Lemma 3.3 since α _i , the internal angle of $w_i= \iota (\zeta _i)=\iota (\psi (2e^{2 \pi i u_i}))$, is just the image φ(u _i ). □

Remark 5.8

The set X′ of all rays landing on ∂Δ is closed and m _d -invariant, and every forward orbit in it has the combinatorial structure of an orbit under r _θ . Yet X′ may fail to be a rotation set. For example, the cubic polynomial

$$\displaystyle \begin{aligned} f(z)=e^{\pi i (\sqrt{5}-1)} z + a z^2 +z^3 \quad \text{with} \quad a \approx 0.44437107-0.35184284\, i \end{aligned}$$

has both critical points c, c′ on ∂Δ with f(c′) = c as shown in Fig. 5.3 left. The critical point c′ is the landing point of four rays at angles $t, t+\frac {1}{9}, t+\frac {1}{3}, t+\frac {4}{9}$ which map under f to the two rays at angles $3t, 3t+\frac {1}{3}$ landing at c ₁. Here t ≈ 0.30762195. The set X′ in this example is not a rotation set since the complement of these six rays already fails to contain two disjoint open intervals of length $\frac {1}{3}$ (Corollary 2.16). However, removing $t+\frac {1}{9}, t+\frac {1}{3}$ and all their preimages from X′ will yield a minimal rotation set X.

Fig. 5.3

Left: Filled Julia set of the cubic map f in Remark 5.8 with both critical points c, c′ on the boundary of the Siegel disk Δ in the center of the picture, where f(c′) = c. Right: A small perturbation of f in Remark 5.12 for which $c' \mapsto c^{\prime }_1=f(c') \mapsto c^{\prime }_2=f^{\circ 2}(c') \in \varDelta$

Remark 5.9

The congruences in (5.5) determine σ(X) uniquely from the knowledge of the internal angles α ₁, …, α _d−1 except when $\alpha _i=0 \ (\operatorname {mod} \ {\mathbb Z})$ for all i. This corresponds to the case where there is a single marked root w ₀ = ⋯ = w _d−2 which is necessarily a critical point of local degree d (compare Corollary 5.11 below). This type of ambiguity has already been pointed out in Remark 3.4 and can now be understood from the dynamical standpoint. For example, when d = 4 and $\alpha _1=\alpha _2=\alpha _3=0 \ (\operatorname {mod} \ {\mathbb Z})$, we have the possible candidates

$$\displaystyle \begin{aligned} \sigma(X)=(0,0,1) \quad \text{or} \quad (0,1,1) \quad \text{or} \quad (1,1,1) \end{aligned}$$

which correspond to quartic polynomials which are conjugate by the 120^∘ rotation around the origin. Dynamically, these cases can be distinguished by the position of the Siegel disk Δ among the three fixed rays $R(0), R(\frac {1}{3}), R(\frac {2}{3})$ (see Fig. 5.4).

Fig. 5.4

Filled Julia set of a unicritical quartic polynomial f(z) = z ⁴ + c with a Siegel disk Δ of the golden mean rotation number. Here the corresponding rotation set X has σ(X) = (0, 0, 1). Conjugating f with the 120^∘ and 240^∘ rotations around the origin yields quartics with σ(X) = (1, 1, 1) and (0, 1, 1). In this example, c ≈ 0.59612528 − 0.46108628 i and ω ≈ 0.68914956

Let us collect some corollaries of Theorem 5.7. As before, let w _i  = ι(ζ _i ) (0 ≤ i ≤ d − 2) be the marked roots of f. To simplify the notation, we denote the limb L(w _i ) by L _i , the wake W(w _i ) by W _i and the gap I(w _i ) by I _i . The following can be thought of as the irrational counterpart of a result of Goldberg and Milnor in [12]:

Theorem 5.10

Let X be the minimal rotation set of Theorem 5.7.

• (i) I ₀, …, I _d−2 are the major gaps of X.

• (ii) The multiplicity n _i of I _i is the number of fixed rays in W _i . It is also the number of subscripts 0 ≤ j ≤ d − 2 for which w _j  = w _i .

• (iii) The limb $L_i=\overline {W_i} \cap K(f)$ contains n _i critical points of f counting multiplicities.

Proof

By the proof of Theorem 5.7 every I _i is a gap of X. Since W _i contains the fixed ray R(u _i ), the gap I _i contains the fixed point u _i of m _d , so it must be major. By Lemma 2.13, the multiplicity n _i of I _i is the number of fixed rays in W _i or the number of times w _i appears in the list w ₀, …, w _d−2. Since there are d − 1 fixed rays, the sum $\sum n_i$ over distinct I _i ’s is d − 1 so I ₀, …, I _d−2 account for all major gaps of X by Theorem 2.7. This proves (i) and (ii).

The proof of (iii) is based on an idea of [12]. Let I _i  = (t, t′), so W _i is bounded by the rays R(t) and R(t′). Let η be a small loop around w _i which intersects each of R(t) and R(t′) once, say at ψ(r ₁e ^2πit) and $\psi (r_1 e^{2 \pi i t'})$. Fix a large radius r ₂. Construct a positively oriented Jordan curve by going out along R(t) from ψ(r ₁e ^2πit) to ψ(r ₂e ^2πit), then following the equipotential curve {ψ(r ₂e ^2πis) : t ≤ s ≤ t′}, then going down along R(t′) from $\psi (r_2 e^{2 \pi i t'})$ to $\psi (r_1 e^{2 \pi i t'})$, and finally going counter-clockwise along η from $\psi (r_1 e^{2 \pi i t'})$ back to ψ(r ₁e ^2πit). Round off the four corners of this curve to obtain a smooth positively oriented Jordan curve γ. The number of the critical points of f in W _i is the number of roots of f′ inside γ. By the argument principle, this is the winding number of the closed curve f′∘ γ around 0, which is one less than the number of full counter-clockwise turns that the tangent vector to image curve f ∘ γ makes when γ is traversed once. By the construction of γ, this number is at least n _i  − k _i  + 1, where k _i  ≥ 1 is the local degree of f at w _i . Taking into account the fact that w _i itself is a critical point of multiplicity k _i  − 1 if k _i  > 1, it follows that the number N _i of the critical points of f in the limb L _i is at least (n _i  − k _i  + 1) + (k _i  − 1) = n _i . Since the sums $\sum N_i$ and $\sum n_i$ over distinct I _i ’s are d − 1, it follows that N _i  = n _i for all i, as required. □

Corollary 5.11

• (i) Every critical point c ∈ ∂Δ is a marked root. Moreover, the algebraic multiplicity of c (as a root of f′) is at most the multiplicity of the corresponding gap I(c).

• (ii) Every marked root w _i whose corresponding gap I _i  = I(w _i ) is taut must be a critical point.

• (iii) A point on ∂Δ is a root if and only if it is pre-critical.

Proof

First suppose c ∈ ∂Δ is a critical point. By Corollary 5.5 the critical value f(c) is the landing point of at least one ray R(t). As in the proof of Lemma 5.3, take small neighborhoods U of c and U′ of f(c) such that f : U → U′ acts as the power w↦w ^k for some k ≥ 2. The intersection R(t) ∩ U′ pulls back under f to the intersection of k rays R(t ₁), …, R(t _k ) with U, all landing at c, where t ₁, …, t _k are among the d preimages of t under m _d . This proves that $K(f) \smallsetminus \{ c \}$ is disconnected, hence c is a root by Lemma 5.2. Moreover, the wake W(c) contains all R(t _i )’s in its closure, so |I(c)|≥ (k − 1)∕d. Hence I(c) is a major gap of X, and the root c is marked by Theorem 5.10(i). The multiplicity n of I(c) is the integer part of d |I(c)|, so n ≥ k − 1. (Alternatively, we could invoke Theorem 5.10(iii) to conclude that n ≥ k − 1.) This proves (i).

To verify (ii), suppose I _i is a taut gap of the form (t, t′ = t + n _i ∕d). Then w _i is the landing point of the rays R(t), R(t′). Under f, these rays map to the same ray R(m _d (t)) = R(m _d (t′)) landing at f(w _i ). This shows f is not injective in any neighborhood of w _i , which proves w _i is a critical point.

For (iii), first note that by part (i) and the backward invariance in Lemma 5.3, all precritical points on ∂Δ are roots. Conversely, consider any root p so I(p) is a gap of the minimal rotation set X of Theorem 5.7. Since ρ(X) is irrational, Theorem 2.10 shows that there is a k ≥ 0 such that $g_X^{\circ k}(I(p))=I(f^{\circ k}(p))$ is a taut gap. By part (ii), f ^∘k(p) is a critical point. □

Remark 5.12

Here are three comments related to various parts of the above corollary: (i) The algebraic multiplicity of a critical point c ∈ ∂Δ can be strictly less than the multiplicity of the gap I(c). This happens precisely when the wake W(c) contains a critical point of f. (ii) If a marked root w _i is critical, the gap I _i may be loose. For example, the cubic map f in Remark 5.8 has both critical points c, c′ on ∂Δ with f(c′) = c, where I(c) is taut and I(c′) is loose (see Fig. 5.3 left). (iii) Marked roots can be non-critical. For example, one can perturb the above map to obtain a cubic with c ∈ ∂Δ and f ^∘2(c′) ∈ Δ (thus the critical point c′ is “captured” by the Siegel disk Δ). Here the second marked root f ⁻¹(c) ∩ ∂Δ is non-critical. Figure 5.3 right shows one such perturbation where

$$\displaystyle \begin{aligned} f(z)=e^{\pi i (\sqrt{5}-1)} z + a z^2 +z^3 \quad \text{with} \quad a \approx 0.54716981 - 0.31132075\, i. \end{aligned}$$

The two examples before and after perturbation have identical minimal rotation sets X. We will discuss this phenomena in more detail in Sect. 5.4.

Corollary 5.13

Suppose all critical points of f are on ∂Δ. Then these critical points are precisely the marked roots w ₀, …, w _d−2, and the algebraic multiplicity of each w _i is equal to the multiplicity of its corresponding gap.

Proof

By Corollary 5.11 all critical points of f are marked roots. Let c ₁, …, c _k be the distinct critical points of multiplicities α ₁, …, α _k . Let n ₁, …, n _k be the multiplicities of the corresponding gaps. By Corollary 5.11(i), α _i  ≤ n _i for all i. Hence, by Theorem 2.7, $d-1 = \sum \alpha _i \leq \sum n_i \leq d-1$. It follows that α _i  = n _i for all i and {c ₁, …, c _k } = {w ₀, …, w _d−1}. □

It would be interesting to investigate how the preceding constructions should be modified for indifferent fixed points with arbitrary irrational rotation numbers. The difficulty arises when the fixed point 0 is the center of a “wild” Siegel disk or is non-linearizable (a so-called “Ceremer point”). In this case, the natural candidate for the rotation set X would be the minimal set of angles of dynamic rays whose impressions meet ∂Δ in the Siegel case and the fixed point 0 in the Cremer case. But in the absence of some kind of control on the Julia set of such maps, proving analogous results seems out of reach even for quadratic polynomials.

5.3 The Quadratic Family

This section and the next illustrate the relation between indifferent fixed points and rotation sets in the low-degree cases d = 2 and d = 3, in both dynamical and parameter planes. The abstract analyses of these rotation sets, carried out in Sects. 4.5 and 4.6 , come to life in these concrete realizations.

The case d = 2 is more straightforward and rather well-known. Consider the monic quadratic polynomial

$$\displaystyle \begin{aligned} P=P_\theta : z \mapsto e^{2 \pi i \theta} z + z^2 \end{aligned} $$

(5.6)

with an indifferent fixed point at the origin. When θ is rational of the form p∕q ≠ 0 in lowest terms, the parabolic fixed point 0 is the landing point of precisely q rays R(t ₁), …, R(t _q ), where X _p∕q = {t ₁, …, t _q } is the unique minimal rotation set under doubling with rotation number p∕q. If as usual we assume 0, t ₁, …, t _q are in positive cyclic order, it follows that the unique critical point c = −e ^2πiθ∕2 lies in the wake bounded by R(t ₁), R(t _q ), corresponding to the longest gap of X _p∕q. Similarly, the critical value v = P(c) = −e ^4πiθ∕4 lies in the wake bounded by R(t _1+p), R(t _q+p), corresponding to the shortest gap of X _p∕q (compare Fig. 5.5 left).

Fig. 5.5

Filled Julia set of the quadratic polynomial z↦e ^2πiθz + z ² with the corresponding minimal rotation set X _θ under doubling. Left: The parabolic case $\theta = \frac {1}{3}$. Right: The Siegel case $\theta = \frac {( \sqrt {5}-1)}{2}$. Shown here are the wakes rooted at the critical point c and its first five preimages on ∂Δ, which define the major gap (ω′, ω) of X _θ and the five minor gaps of lengths 2^−k for 2 ≤ k ≤ 6

When θ is an irrational of bounded type (or more generally belongs to the full-measure set $\mathcal E$ in [25]), the Julia set J(P) is locally connected. In this case the boundary of the Siegel disk Δ of P centered at 0 is a Jordan curve containing c, and the limb decomposition hypothesis automatically holds. It follows from the general results of the previous section that the set of angles of the rays that land on ∂Δ is precisely the minimal rotation set X _θ under doubling. Note that X _θ is a Cantor set with a single major gap of length $\frac {1}{2}$ bounded by the angles $\omega , \omega '=\omega -\frac {1}{2}$, where $0<\omega = \omega (\theta )<\frac {1}{2}$ is the leading angle of X _θ as defined in Sect. 4.5 . By Corollary 5.11, both rays R(ω), R(ω′) land at the critical point c which is the unique marked root. The precritical point P ⁻ⁿ(c) ∩ ∂Δ is then the root whose corresponding wake defines the gap of X _θ of length $\frac {1}{2^n}$ (see Fig. 5.5 right).

The realization of rotation sets in the dynamical plane allows an alternative route to Lemma 4.24 . The binary expansion 0.b ₀b ₁b ₂⋯ of the leading angle ω = ω(θ) of X _θ is characterized by the condition b _k  = 1 if and only if $2^k \omega \in (\frac {1}{2},1)$. If θ = p∕q ≠ 0 and X _p∕q = {t ₁, …, t _q } as above, then 0 ∈ (t _q , t ₁) and $\frac {1}{2} \in (t_{q-p},t_{q-p+1})$. Hence $t_1, \ldots , t_{q-p} \in (0,\frac {1}{2})$ while $t_{q-p+1}, \ldots , t_q \in (\frac {1}{2},1)$. Thus,

$$\displaystyle \begin{aligned} 2^k \omega = t_{1+kp} \in \Big(\frac{1}{2},1\Big) \ \Longleftrightarrow \ 1+kp \ (\text{mod} \ q) \ \text{is in} \ \{ q-p+1, \ldots, q \}.\end{aligned} $$

This is clearly equivalent to kθ ∈ [−θ, 0).

A similar argument works when θ is an irrational and P has a “good” Siegel disk. In this case, 0 ∈ (ω′, ω) and $\frac {1}{2} \in ((\omega '+1)/2,(\omega +1)/2)$, so $2^k \omega \in (\frac {1}{2},1)$ if and only if 2^kω ∈ ((ω + 1)∕2, ω′). But the pair R(ω), R(ω′) land at c with the internal angle 0 and the pair R((ω + 1)∕2), R((ω′ + 1)∕2) land at the preimage P ⁻¹(c) ∩ ∂Δ with the internal angle − θ. It follows that 2^kω ∈ ((ω + 1)∕2, ω′) precisely when kθ, the internal angle of P ^∘k(c), is in the interval (−θ, 0).

The parameter space of quadratic polynomials provides a complete catalog of all rotation sets under doubling. To see this, it will be convenient to represent our quadratics in the normal form f _c (z) = z ² + c where $c \in {\mathbb C}$. The connectedness locus

$$\displaystyle \begin{aligned} \mathbb{M}_2 = \{ c \in {\mathbb C}: K(f_c) \ \text{is connected} \},\end{aligned} $$

commonly known as the Mandelbrot set , is non-empty, compact, and full. If β _c denotes the Böttcher coordinate of f _c near ∞, the Douady-Hubbard map $\varPhi : {\mathbb C} \smallsetminus \mathbb {M}_2 \to {\mathbb C} \smallsetminus \overline {{\mathbb D}}$ which assigns to each c outside $\mathbb {M}_2$ the Böttcher coordinate β _c (c) of the critical value f _c (0) = c, is a conformal isomorphism. By the parameter ray of $\mathbb {M}_2$ at angle $t \in {\mathbb T}$ we mean the real-analytic curve

$$\displaystyle \begin{aligned} {\mathbb{R}}(t)= \varPhi^{-1} \big( \{ r e^{2 \pi i t} : r>1 \} \big).\end{aligned} $$

We say $\mathbb {R}(t)$ lands at $z \in \partial \mathbb {M}_2$ if lim_r→1Φ ⁻¹(re ^2πit) = z.

Each quadratic P _θ in (5.6) is affinely conjugate to f _c with c = c(θ) = e ^2πiθ∕2 − e ^4πiθ∕4. As θ varies in [0, 1], the image c(θ) traces out a cardioid on the boundary of $\mathbb {M}_2$ that is prominently visible in Fig. 5.6. When θ ≠ 0 is rational, c(θ) is the landing point of the two parameter rays $\mathbb {R}(2\omega ), \mathbb {R}(2\omega ')$. (Recall that (ω′, ω) is the major gap of X _θ .) If θ is irrational, then c(θ) is the landing point of the unique parameter ray $\mathbb {R}(2\omega )=\mathbb {R}(2\omega ')$. One may interpret this by saying that c(θ) is always the landing point of the parameter ray at angle 2ω(θ), which is a strictly increasing function of θ that jumps by 1∕(2^q − 1) at every rational θ = p∕q (Corollary 4.26 ). When θ is rational, the two parameter rays $\mathbb {R}(2\omega ), \mathbb {R}(2\omega ')$ together with their landing point c(θ) define the parameter wake $\mathbb {W}(\theta )$, characterized by the property that the dynamic rays with angles in X _θ land at a fixed point of f _c if and only if $c \in \mathbb {W}(\theta ) \cap \mathbb {M}_2$ (for a detailed treatment see [20] and compare Fig. 5.6).

Fig. 5.6

The Mandelbrot set $\mathbb {M}_2$ and its parameter wakes $\mathbb {W}( \frac {1}{3})$, $\mathbb {W}( \frac {1}{2})$ and $\mathbb {W}( \frac {2}{3})$. Also shown is the parameter ray $\mathbb {R}(2\omega )$ landing at the quadratic that is affinely conjugate to e ^2πiθz + z ². Here $\theta = \frac {( \sqrt {5}-1)}{2}$ and ω = ω(θ) ≈ 0.35490172

Remark 5.14

The family of degree d unicritical polynomials z↦z ^d + c exhibits very similar features in relation with rotation sets. As an example, the cubic map f _c  : z↦z ³ + c has an indifferent fixed point of multiplier e ^2πiθ if and only if

$$\displaystyle \begin{aligned} c=\pm c(\theta) \quad \text{where} \quad c(\theta)= - \frac{1}{3\sqrt{3}} \, e^{3 \pi i \theta} + \frac{1}{\sqrt{3}} \, e^{\pi i \theta}. \end{aligned}$$

The maps f _c(θ) and f _−c(θ) are conjugate by the 180^∘ rotation z↦ − z. The angles of the dynamic rays of f _c(θ) that land on the indifferent fixed point when θ is rational, or on the boundary of the Siegel disk when θ is a suitable irrational, form the rotation set X _θ,1 under tripling. The rotation set associated with the conjugate map f _−c(θ) is of course X _θ,0. As θ varies in [0, 1], the images ± c(θ) trace out an algebraic curve (a nephroid) on the boundary of the corresponding connectedness locus $\mathbb {M}_3$ which bounds the central hyperbolic component containing c = 0. The analog of the Douady-Hubbard map is a conformal isomorphism ${\mathbb C} \smallsetminus \mathbb {M}_3 \to {\mathbb C} \smallsetminus \overline {{\mathbb D}}$, which can be used to define parameter rays in the c-plane. The boundary point c(θ) is the landing point of the parameter ray at angle 3ω(θ, 1), which strictly increases from 0 to $\frac {1}{2}$, jumping by 2∕(3^q − 1) at every rational θ = p∕q (Corollary 4.32 ). Similarly, − c(θ) is the landing point of the parameter ray at angle $3 \omega (\theta ,0)=3 \omega (\theta ,1)+\frac {1}{2}$, which strictly increases from $\frac {1}{2}$ to 1 with similar jumps at every rational θ. As in the case of the Mandelbrot set, there is an analogous notion of parameter wakes for $\mathbb {M}_3$ and their dynamical characterization (see Fig. 5.7).

Fig. 5.7

The connectedness locus $\mathbb {M}_3$ of the unicritical cubic family $\{ f_c: z \mapsto z^3+c \}_{c \in { \mathbb C}}$, with selected parameter rays and wakes. Here $\mathbb {W}(p/q,\delta ) \cap \mathbb {M}_3$ for δ = 0, 1 is precisely the set of parameters c for which the dynamical rays at angles in X _p∕q,δ land at a fixed point of f _c

5.4 The Cubic Family

This section is somewhat expository and contains outlines of the results. Consider the space of monic cubic polynomials with an indifferent fixed point of multiplier e ^2πiθ at the origin. Each such cubic has the form

$$\displaystyle \begin{aligned} f_a : z \mapsto e^{2 \pi i \theta} z + a z^2 + z^3 \quad \text{for some} \quad a \in {\mathbb C}. \end{aligned} $$

(5.7)

Note that f _a and f _−a are affinely conjugate by the involution z↦ − z. One could thus look at the quotient of the a-plane under a↦ − a (equivalently, work with the parameter a ²). However, for our purposes in this section we prefer to treat f _a and f _−a as distinct cubics.

The connectedness locus of this cubic family is defined by

$$\displaystyle \begin{aligned} \mathbb{M}_3(\theta) = \{ a \in {\mathbb C}: K(f_a) \ \text{is connected} \}. \end{aligned}$$

It is not hard to verify that $\mathbb {M}_3(\theta )$ is a compact, connected and full subset of ${\mathbb C}$ which is invariant under the involution a↦ − a [30].

When $a \in \mathbb {M}_3(\theta )$, both critical points of f _a belong to the filled Julia set K(f _a ). When $a \notin \mathbb {M}_3(\theta )$, exactly one of the critical points, labeled c _a , belongs to K(f _a ) while the other, labeled e _a , escapes to ∞. The escaping critical value v _a  = f _a (e _a ) has two preimages under f _a : the critical point e _a itself (with multiplicity 2) and a regular point $\hat {e}_a$ which we call the escaping co-critical point . The Böttcher coordinate β _a of f _a near ∞ is defined and holomorphic in some neighborhood of $\hat {e}_a$. The analog of the Douady-Hubbard map $\varPhi : {\mathbb C} \smallsetminus \mathbb {M}_3(\theta ) \to {\mathbb C} \smallsetminus \overline {{\mathbb D}}$ defined by

$$\displaystyle \begin{aligned} \varPhi(a) = \beta_a(\hat{e}_a) \end{aligned}$$

is a conformal isomorphism [6]. We define the parameter ray at angle $t \in {\mathbb T}$ by

$$\displaystyle \begin{aligned} \mathbb{R}(t)= \{ \varPhi^{-1}(re^{2\pi i t}) : r>1 \}. \end{aligned}$$

We study the realization of rotation sets under m ₃ in the dynamical plane of f _a as well as the parameter a-plane. The discussion is presented in two cases depending on whether θ is rational or an irrational of bounded type. We will outline the first case only briefly, as our main interest is the case of cubics with Siegel disks.

The parabolic case. Let us assume θ is rational of the form p∕q ≠ 0 in lowest terms. By the discussion of Sect. 5.2, the q-th iterate of f _a has the form

$$\displaystyle \begin{aligned} f_a^{\circ q}(z) = z + A(a) \, z^{q+1} + \cdots+ z^{3^q}. \end{aligned}$$

Here A(a) is a polynomial of degree q in a with simple roots. Moreover, A is an even function if q is even, and odd function if q is odd. If A(a) ≠ 0, the petal number of the parabolic point 0 is q and its ray number is q or 2q. If, on the other hand, A(a) = 0, then the above expression reduces to

$$\displaystyle \begin{aligned} f_a^{\circ q}(z) = z + B(a) \, z^{2q+1} + \cdots+ z^{3^q}. \end{aligned}$$

where B(a) ≠ 0, so the petal and ray numbers are both 2q. In this case, we say f _a has a degenerate parabolic fixed point at 0.

By Lemma 5.1 the set X _a of angles of the dynamic rays of f _a that land at 0 is a rotation set under tripling with ρ(X _a ) = p∕q, which consists of one or two q-cycles. The deployment vector of X _a has the form δ(X _a ) = (δ _a , 1 − δ _a ), where δ _a  ∈ [0, 1] is the deployment probability of f _a , i.e., the probability that a dynamic ray R _a (t) of f _a landing on 0 has its angle t in $(0,\frac {1}{2})$. Note that by symmetry,

$$\displaystyle \begin{aligned} \delta_{-a} = 1-\delta_a \qquad a \in \mathbb{M}_3(p/q). \end{aligned}$$

First suppose the ray number is q, so X _a is a single q-cycle {t ₁, …, t _q }. Thus, in the notation of Sect. 4.6 , X _a  = X _p∕q,i∕q for some 0 ≤ i ≤ q. If we assume 0, t ₁, …, t _q are in positive cyclic order, it follows that one critical point of f _a lies in the wake bounded by the dynamic rays R _a (t _q ), R _a (t ₁), the other in the wake bounded by R _a (t _i ), R _a (t _i+1). Thus, the deployment probability δ _a  = i∕q is determined by the “combinatorial distance” i between the two critical points of f _a (that is, how many wakes they are apart). Figure 5.1 left illustrates this case with $p/q=i/q=\frac {1}{3}$.

Next consider the case where the ray number is 2q, so X _a  = {t ₁, …, t _2q}. Under tripling, each t _j maps to t _j+2p so X _a splits into two q-cycles. As these q-cycles are compatible, Theorem 3.16 shows that

$$\displaystyle \begin{aligned} X_a = X_{p/q,i/q} \cup X_{p/q,(i+1)/q}\end{aligned} $$

for some 0 ≤ i ≤ q − 1. Now one critical point of f _a lies in the wake bounded by R _a (t _2q), R _a (t ₁), the other in the wake bounded by R _a (t _2i+1), R _a (t _2i+2). Thus, similar to the above case, the deployment probability δ _a  = (2i + 1)∕(2q) is determined by the combinatorial distance 2i + 1 between the two critical points of f _a . Figure 5.1 right illustrates this case with $p/q=i/q=\frac {1}{3}$.

Turning the attention to the parameter space, one can identify the following types of the interior components for $\mathbb {M}_3(p/q)$:

• adjacent, where the two critical points belong to the same attracting petal at 0;

• bi-transitive, where the two critical points belong to different attracting petals at 0 in the same cycle;

• capture, where the orbit of one critical point eventually hits the cycle of attracting petals at 0;

• hyperbolic-like, where the orbit of one critical point converges to an attracting cycle.

Conjecturally, every interior component of $\mathbb {M}_3(p/q)$ is of one of the above types. In fact, the only possibility to rule out is a “queer” component in a small copy of the Mandelbrot set in $\mathbb {M}_3(p/q)$ in which the interior of K(f _a ) is the basin of attraction of 0 but the Julia set J(f _a ) has positive measure and admits an invariant line field.

Let a ₀, …, a _q−1 denote the degenerate parabolic parameters, i.e., simple roots of the equation A(a) = 0. There is a chain of interior components C ₀, C ₁, …, C _q of $\mathbb {M}_3(p/q)$ such that ∂C _i−1 ∩ ∂C _i  = {a _i } for 1 ≤ i ≤ q. Here C _i  = −C _q−i, with C ₀ and C _q of adjacent type and C ₁, …, C _q−1 of bi-transitive type (see Fig. 5.8). For every parameter a ∈ C _i , we have δ _a  = i∕q.

Fig. 5.8

The parabolic connectedness locus $\mathbb {M}_3( \frac {2}{3})$ and the chain of interior components C ₀, C ₁, C ₂, C ₃. The twelve parameter rays landing on the degenerate cubics a ₀, a ₁, a ₂ define the ten wakes $\mathbb {W}_0, \mathbb {W}_1, \mathbb {W}_2 \mathbb {W}_3$ and $\varOmega _0^\pm , \varOmega _1^\pm , \varOmega _2^\pm$. The deployment probability δ _a takes the value i∕3 on $\mathbb {W}_i \cap \mathbb {M}_3( \frac {2}{3})$ and (2i + 1)∕6 on $\varOmega _i^\pm \cap \mathbb {M}_3( \frac {2}{3})$

The deployment probability δ _a can be determined throughout the connectedness locus $\mathbb {M}_3(p/q)$. Each degenerate parabolic parameter a _i is the landing point of four parameter rays whose angles are those of the dynamic rays of $f_{a_i}$ that bound the Fatou components containing its co-critical points. Using the general results of Sect. 4.6 it is not hard to find explicit formulas for these angles in terms of the leading angles ω(p∕q, i∕q) and ω(p∕q, (i + 1)∕q). An example of this computation for $p/q=\frac {2}{3}$ and i = 0 is shown in Fig. 5.9.

Fig. 5.9

Filled Julia set of the degenerate parabolic f _a in $\mathbb {M}_3( \frac {2}{3})$ with $X_a=X_{ \frac {2}{3}, \frac {0}{3}} \cup X_{ \frac {2}{3}, \frac {1}{3}} = \{ \frac {24}{78}, \frac {51}{78}, \frac {60}{78}, \frac {69}{78}, \frac {72}{78}, \frac {75}{78} \}$ and $\delta _a= \frac {1}{6}$. Here a ≈ 0.68308975 − 1.08669099 i. The ray pairs at angles $( \frac {75}{78}, \frac {24}{78})$ and $( \frac {24}{78}, \frac {51}{78})$ bound the Fatou components containing the critical points c and c′, respectively. It follows that the ray pairs at angles $( \frac {75}{78}- \frac {1}{3}= \frac {49}{78}, \frac {24}{78}+ \frac {1}{3}= \frac {50}{78})$ and $( \frac {24}{78}+ \frac {2}{3}= \frac {76}{78}, \frac {51}{78}+ \frac {1}{3}= \frac {77}{78})$ bound the Fatou components containing the co-critical points $\hat {c}$ and $\hat {c}'$, respectively

These 4q parameter rays together with their landing points {a ₀, …, a _q−1} divide the a-plane into 3q + 1 parameter wakes $\mathbb {W}_0, \ldots , \mathbb {W}_q, \varOmega _0^{\pm }, \ldots , \varOmega _{q-1}^{\pm }$. Here $\mathbb {W}_i$ contains C _i and the pair $\varOmega _i^{\pm }$ separate $\mathbb {W}_i$ from $\mathbb {W}_{i+1}$ (see Fig. 5.8). We have X _a  = X _p∕q,i∕q if $a \in \mathbb {W}_i \cap \mathbb {M}_3(p/q)$, and X _a  = X _p∕q,i∕q ∪ X _{p∕q,(i+1)∕q} if $a \in \varOmega _i^{\pm } \cap \mathbb {M}_3(p/q)$. Thus,

$$\displaystyle \begin{aligned} \delta_a= \begin{cases} \frac{i}{q} & \quad \text{if} \ a \in \mathbb{W}_i \cap \mathbb{M}_3(p/q) \vspace{2mm} \\ \frac{2i+1}{2q} & \quad \text{if} \ a \in \varOmega_i^{\pm} \cap \mathbb{M}_3(p/q). \end{cases} \end{aligned}$$

A detailed analysis of the landing properties of some of the parameter rays of $\mathbb {M}_3(p/q)$ can be found in [3].

The “good” Siegel case. Now suppose θ is an irrational of bounded type, so the fixed point 0 of f _a is the center of a Siegel disk Δ _a . The boundary ∂Δ _a is then a Jordan curve (in fact a quasicircle) passing through one or both critical points of f _a .

One can easily identify the following two types of interior components of the connectedness locus $\mathbb {M}_3(\theta )$:

• capture, where the orbit of one critical point eventually hits the Siegel disk;

• hyperbolic-like, where the orbit of one critical point converges to an attracting cycle.

As in the rational case, it is conjectured that every interior component of $\mathbb {M}_3(\theta )$ has one of these types. In Fig. 5.10 left the capture components are the blue bulbs, while the hyperbolic-like components are the grey bulbs that belong to a small copy of the Mandelbrot set.

Fig. 5.10

Left: The cubic connectedness locus $\mathbb {M}_3(\theta ) \subset { \mathbb C}$. Right: The arc $\varGamma (\theta ) \subset \mathbb {M}_3(\theta )$. Here $\theta = \frac {( \sqrt {5}-1)}{2}$

The following is proved in [30]:

Theorem 5.15

There is a closed embedded arc $\varGamma (\theta ) \subset \mathbb {M}_3(\theta )$ with the property that a ∈ Γ(θ) if and only if ∂Δ _a contains both critical points of f _a .

The arc Γ(θ) is clearly invariant under the involution a↦ − a. The endpoints of Γ(θ) are the parameters $\pm \sqrt {3 e^{2\pi i \theta }}$ corresponding to the cubics with a double critical point. We denote by a ₀ the endpoint in the lower half-plane, so − a ₀ is the other endpoint in the upper half-plane. The midpoint of Γ(θ) is the parameter a = 0 corresponding to the cubic with centered critical points. See Fig. 5.10 right.³

The arc Γ(θ) is parametrized by the internal angle between the two critical points (as defined in Sect. 5.2). More precisely, if a ∈ Γ(θ) and if the internal angles of the critical points of f _a are 0 and τ _a  ∈ [0, 1], where $\tau _{a_0}=0$ and $\tau _{-a_0}=1$, then the map a↦τ _a is a homeomorphism Γ(θ) → [0, 1].

Here are two alternative characterizations of Γ(θ):

• Γ(θ) is the set of parameters near which the boundary ∂Δ _a fails to move holomorphically. In fact, if U is a disk which does not intersect Γ(θ), then the critical point of f _a that lies on ∂Δ _a depends holomorphically on a ∈ U, so its forward orbit moves holomorphically over U. By the λ-lemma [16], this holomorphic motion extends to a holomorphic motion of the closure of this forward orbit, which is just ∂Δ _a . On the other hand, if U is a disk that does intersect Γ(θ), the critical point on ∂Δ _a cannot be followed holomorphically in U, which shows ∂Δ _a does not move holomorphically over U (although it still moves continuously in the Hausdorff topology [30]).

• Let $\operatorname {rad}(a)$ denote the conformal radius of the Siegel disk Δ _a relative to its center 0. The function $a \mapsto \log \operatorname {rad}(a)$ is continuous and subharmonic in ${\mathbb C}$ and harmonic off Γ(θ) (see [5] and [32]). The arc Γ(θ) can be described as the support of the generalized Laplacian $4 \partial \overline {\partial } \log \operatorname {rad}$. This has been proved by I. Zidane and independently by the author (unpublished).

An adaptation of the work of Petersen in [23], using complex a priori bounds for critical circle maps, proves that for every a ∈ Γ(θ) the Julia set of f _a is locally connected and has measure zero. Thus, along Γ(θ) the Julia set is tame enough to allow the general constructions of Sect. 5.2 to go through. In particular, it follows from Theorem 5.7 that we can assign to each a ∈ Γ(θ) a minimal rotation set X _a under tripling with ρ(X _a ) = θ, consisting of angles of the dynamic rays of f _a which land on ∂Δ _a . Notice the symmetry

$$\displaystyle \begin{aligned} X_{-a} = X_a + \frac{1}{2} \quad (\operatorname{mod} \ {\mathbb Z}). \end{aligned} $$

(5.8)

For each a ∈ Γ(θ) consider the deployment vector δ(X _a ) = (δ _a , 1 − δ _a ), where δ _a  ∈ [0, 1] is the deployment probability of f _a , i.e., the probability that a dynamic ray R _a (t) landing on ∂Δ _a has its angle t in $(0,\frac {1}{2})$. It follows from the symmetry relation (5.8) that

$$\displaystyle \begin{aligned} \delta_{-a} = 1-\delta_a \qquad a \in \varGamma(\theta). \end{aligned} $$

At the two endpoints a = ±a ₀ of Γ(θ) the cubic f _a has a double critical point whose wake contains both dynamic rays R _a (0) and $R_a(\frac {1}{2})$. At any other a ∈ Γ(θ) the critical points of f _a are distinct and we label them as ∗ _a and $*^{\prime }_a$ by requiring that the wake W(∗ _a ) contains R _a (0) and the wake $W(*^{\prime }_a)$ contains $R_a(\frac {1}{2})$. Under this labeling, the internal angle of ∗ _a will be 0 and that of $*^{\prime }_a$ will be τ _a .

The following is an immediate corollary of Theorem 5.7:

Theorem 5.16

For every parameter a ∈ Γ(θ), the deployment probability of X _a is the internal angle between the two critical points of f _a :

$$\displaystyle \begin{aligned} \delta_a = \tau_a. \end{aligned} $$

Thus, starting at the endpoint a ₀ of Γ(θ) in the lower half-plane and moving to the other endpoint − a ₀, the probability δ _a increases monotonically and takes all values between 0 and 1. In particular, the family {X _a }_{a ∈ Γ(θ)} spans all minimal rotation sets under tripling with ρ(X _a ) = θ.

For each integer n ≥ 1, let a _n be the unique parameter on Γ(θ) for which $\delta _{a_n}=n\theta \ (\operatorname {mod} \ {\mathbb Z})$ (the first few a _n are shown in Fig. 5.10 right). Using Theorem 5.16, it is readily seen that $f_{a_n}^{\circ n}(*_{a_n})=*^{\prime }_{a_n}$. By Theorem 4.31 , the rotation set $X_{a_n}$ has a taut gap of length $\frac {1}{3}$ corresponding to the wake $W(*^{\prime }_{a_n})$ and a loose gap of length $\frac {1}{3}+\frac {1}{3^{n+1}}$ corresponding to the wake $W(*_{a_n})$ (compare Fig. 5.12). Of course by symmetry the parameters − a _n have similar dynamical description, with ∗ _a and $*^{\prime }_a$ exchanged. Namely, $\delta _{-a_n}=-n\theta \ (\operatorname {mod} \ {\mathbb Z})$, $f_{-a_n}^{\circ n}(*^{\prime }_{-a_n})=*_{-a_n}$, and $X_{-a_n}$ has a taut gap of length $\frac {1}{3}$ corresponding to $W(*_{-a_n})$ and a loose gap of length $\frac {1}{3}+\frac {1}{3^{n+1}}$ corresponding to $W(*^{\prime }_{-a_n})$.⁴

We can combinatorially describe Γ(θ) by specifying the angles of the candidate parameter rays that presumably land on it. This description is related to rotation sets under tripling, much like what we have seen in the case of the boundary of the main cardioid of the Mandelbrot set. It will be convenient to use Theorem 5.16 to parametrize Γ(θ) by the deployment probability. For each δ ∈ [0, 1], let a(δ) ∈ Γ(θ) be the unique parameter with δ _a(δ) = δ. Thus, $a(\frac {1}{2})=0$ and in terms of our previous notation, a(0) = a ₀, a(1) = −a ₀, and a(±nθ) = ±a _n for all n ≥ 1. If $\delta \neq n\theta \ (\operatorname {mod} \ {\mathbb Z})$ for all n, there are two angles $-\frac {1}{6}<s(\delta )<\frac {1}{6}$ and $\frac {1}{3}<t(\delta )<\frac {2}{3}$ such that the parameter rays $\mathbb {R}(s(\delta ))$ and $\mathbb {R}(t(\delta ))$ land at a(δ) (thus, in Fig. 5.14, $\mathbb {R}(s(\delta ))$ lands at a(δ) from the right side of Γ(θ) while $\mathbb {R}(t(\delta ))$ lands there from the left side). These angles can be expressed in terms of the leading angle ω(θ, δ) of X _a(δ) = X _θ,δ studied in Sect. 4.6 :

$$\displaystyle \begin{aligned} t(\delta) & = \omega(\theta,\delta)+\frac{1}{3} \\ {} s(\delta) & = \omega(\theta,1-\delta)-\frac{1}{6}\end{aligned} $$

This can be seen by examining Fig. 5.11 which illustrates the angles of the dynamic rays landing at the co-critical points of f _a(δ). Notice that by symmetry,

$$\displaystyle \begin{aligned} t(\delta) = s(1-\delta)+\frac{1}{2}.\end{aligned} $$

Fig. 5.11

Filled Julia set of a typical cubic map f _a with a ∈ Γ(θ), where the critical points ∗, ∗′ have disjoint orbits on ∂Δ. Here the rays at angles $t \pm \frac {1}{3}$ land at ∗ and those at angles $s \pm \frac {1}{3}$ land at ∗′. If δ is the deployment probability of the associated rotation set X _a , we have $t- \frac {1}{3}=\omega (\theta ,\delta )$ and $s- \frac {1}{3}=\omega (\theta ,1-\delta )+ \frac {1}{2}$. Thus, the rays landing at the co-critical points $\hat {*}, \hat {*}'$ have angles $t= \omega (\theta ,\delta )+ \frac {1}{3}$ and $s=\omega (\theta ,1-\delta )- \frac {1}{6}$, respectively

Fig. 5.12

Filled Julia set of the cubic map $f_{a_n}$, where the n-th iterate of the critical point ∗ hits the critical point ∗′. Here the rays at angles $s \pm \frac {1}{3}$ land at ∗′ and those at angles $t \pm \frac {1}{3}$ and $t \pm \frac {1}{3}- \frac {1}{3^{n+1}}$ land at ∗ (although only two of them, shown in the picture, are present in the rotation set $X_{a_n}$). We have $t- \frac {1}{3}=\omega (\theta ,n\theta )$ and $s- \frac {1}{3}=\omega (\theta ,-n\theta )+ \frac {1}{2}$. Thus, the ray at angle $s = \omega (\theta ,-n\theta )- \frac {1}{6}$ lands at the co-critical point $\hat {*}'$ and the rays at angles $t=\omega (\theta ,n\theta )+ \frac {1}{3}$ and $t- \frac {1}{3^{n+1}}=\omega (\theta ,n\theta )+ \frac {1}{3}- \frac {1}{3^{n+1}}$ land at the co-critical points $\hat {*}$

Recall from Theorem 4.33 that the leading angle δ↦ω(θ, δ) is a decreasing, left-continuous function with a jump discontinuity of size $\frac {1}{3^{n+1}}$ at $\delta =n \theta \ (\operatorname {mod} \ {\mathbb Z})$ for each n ≥ 0. Moreover,

$$\displaystyle \begin{aligned} \omega(\theta,0)=\omega(\theta,0^+)+\frac{1}{3}=\omega(\theta,1)+\frac{1}{2}.\end{aligned} $$

It follows from the above formulas that s(δ) is increasing and t(δ) is decreasing as a function of δ. For each n ≥ 1 the angle t(δ) has a jump discontinuity of size $\frac {1}{3^{n+1}}$ at $\delta =n \theta \ (\operatorname {mod} \ {\mathbb Z})$, while s(δ) remains continuous there, and similarly, s(δ) has a jump discontinuity of size $\frac {1}{3^{n+1}}$ at $\delta =-n \theta \ (\operatorname {mod} \ {\mathbb Z})$, while t(δ) remains continuous there. These values of δ correspond to the parameters ± a _n along Γ(θ) and the aforementioned discontinuity suggests that every a _n with n ≥ 1 is the landing point of three parameter rays at angles

$$\displaystyle \begin{aligned} t_n^- & = \omega(\theta,n \theta)+\frac{1}{3}-\frac{1}{3^{n+1}} \\ t_n^+ & = \omega(\theta,n \theta)+\frac{1}{3} \\ s_n & = \omega(\theta,-n\theta)-\frac{1}{6}\end{aligned} $$

while the parameter − a _n is the landing point of the three parameter rays at angles

$$\displaystyle \begin{aligned} s_n^- & = \omega(\theta,n \theta)-\frac{1}{6}-\frac{1}{3^{n+1}} \\ s_n^+ & = \omega(\theta,n\theta)-\frac{1}{6}\\ t_n & = \omega(\theta,-n\theta)+\frac{1}{3}.\end{aligned} $$

These computations are illustrated in Fig. 5.12 which shows the angles of the dynamic rays that land at the co-critical points of $f_{a_n}$.

Finally, the endpoint a ₀ of Γ(θ) is the landing point of the two parameter rays at angles

$$\displaystyle \begin{aligned} t_0^- & = \omega(\theta,1) + \frac{1}{2} \\ t_0^+ & = \omega(\theta,1) + \frac{5}{6},\end{aligned} $$

while the other endpoint − a ₀ is the landing point of the two parameter rays at angles

$$\displaystyle \begin{aligned} s_0^- & = \omega(\theta,1) \\ s_0^+ & = \omega(\theta,1) + \frac{1}{3}.\end{aligned} $$

Compare Fig. 5.13 which provides a justification for these formulas.

Fig. 5.13

Filled Julia set of the cubic map $f_{a_0}$ with a double critical point ∗ = ∗′ (which also coincides with the co-critical points $\hat {*}=\hat {*}'$). Here the rays at angles $t=\omega (\theta ,1)+ \frac {5}{6}$ and $t- \frac {1}{3}=\omega (\theta ,1)+ \frac {1}{2}$ land at ∗

By Theorem 4.35 , the above angles can be expressed rationally in terms of the (transcendental) base angle ω = ω(θ, 1). It follows that

$$\displaystyle \begin{aligned} t_n^+ & = \frac{(3^n+1) \omega + A_n}{2 \cdot 3^n} +\frac{1}{3}\\ s_n & = \frac{(3^n+1) \omega - B_n}{2} - \frac{1}{6}, \end{aligned} $$

where A _n , B _n are the integers defined by (4.16 ).

Example 5.17

For the golden mean $\theta =\frac {(\sqrt {5}-1)}{2}$, the base angle ω = ω(θ, 1) can be effectively computed with desired precision using the formula (4.13 ):

$$\displaystyle \begin{aligned} \omega \approx 0.128099593431 \cdots \end{aligned}$$

Using the formula (4.16 ) it is easy to compute the integers A _n , B _n . Here are the results for 1 ≤ n ≤ 5:

$$\displaystyle \begin{aligned} A_1 & = 3^0=1 & & & B_1 & = 0 \\ A_2 & = 3^0+3^1=4 & & & B_2 & = 3^0=1 \\ A_3 & = 3^0+3^1=4 & & & B_3 & = 3^1=3 \\ A_4 & = 3^0+3^1+3^3=31 & & & B_4 & = 3^0+3^2=10 \\ A_5 & = 3^0+3^1+3^3+3^4=112 & & & B_5 & = 3^0+3^1+3^3=31. \\ \end{aligned} $$

The corresponding angles are listed in Table 5.1. Figure 5.14 shows selected parameter rays at these angles.

Table 5.1 Angles of some parameter rays which “land” on the arc Γ(θ) for $\theta = \frac {( \sqrt {5}-1)}{2}$

Fig. 5.14

Some parameter rays which “land” on the roots of capture components along the arc Γ(θ). Here $\theta = \frac {( \sqrt {5}-1)}{2}$

We can extend this picture to parameters outside the arc Γ(θ). One possible approach is to show that when θ is of bounded type, the filled Julia sets K(f _a ) for $a \in \mathbb {M}_3(\theta )$ satisfy the limb decomposition hypothesis in Sect. 5.2 so the rotation set X _a is well defined. This is already known for many parameters in $\mathbb {M}_3(\theta )$, including the hyperbolic-like ones, and is surely true for all capture parameters. An alternative route, which is outlined below, is to approach $\mathbb {M}_3(\theta )$ from outside, allowing disconnected Julia sets.

Outside the connectedness locus, the filled Julia set K(f _a ) consists of countably many homeomorphic copies of the filled Julia set of the quadratic polynomial P : z↦e ^2πiθz + z ² and uncountably many points. In particular, the connected component K _a of K(f _a ) containing the Siegel disk Δ _a , called the little filled Julia set , is homeomorphic to K(P). More precisely, let $G_a:{\mathbb C} \to {\mathbb R}$ be the Green’s function of f _a as defined in Sect. 5.1, and U _a and V _a be the connected components of $G_a^{-1}[0,G_a(e_a))$ and $G_a^{-1}[0,G_a(e_a)/3)$ containing K _a , respectively (recall that e _a is the escaping critical point). Then U _a and V _a are Jordan domains with $K_a \subset V_a \subset \overline {V_a} \subset U_a$ and the restriction f _a  : V _a  → U _a is a degree 2 branched covering (see Fig. 5.15). According to Douady and Hubbard, this restriction is hybrid equivalent to the quadratic P, namely, there is a quasiconformal homeomorphism ϕ _a  : U _a  → ϕ _a (U _a ) which satisfies ϕ _a  ∘ f _a  = P ∘ ϕ _a in V _a , with ϕ _a (K _a ) = K(P) and $\overline {\partial } \phi _a=0$ a. e. on K _a (see for example [30] or [6]).

Fig. 5.15

Filled Julia set of a cubic f _a outside the connectedness locus $\mathbb {M}_3(\theta )$. The restriction f _a  : V _a  → U _a is a degree 2 branched covering hybrid equivalent to the quadratic z↦e ^2πiθz + z ²

When a is outside $\mathbb {M}_3(\theta )$, it belongs to the parameter ray $\mathbb {R}(t)$ for a unique $t \in {\mathbb T}$ called the external angle of a. It follows that the dynamic rays $R_a(t \pm \frac {1}{3})$ are bifurcated and crash into the escaping critical point e _a . Let N _t be the countable dense set of angles whose forward m ₃-orbit hit either of $t \pm \frac {1}{3}$. If u∉N _t , the ray R _a (u) is smooth. If u ∈ N _t , the ray R _a (u) is bifurcated and crashes into an iterated preimage of e _a (only once if neither $t \pm \frac {1}{3}$ is periodic under m ₃, infinitely many times otherwise). For each u ∈ N _t we can define the limit rays R _a (u ^±) as the pointwise limits

$$\displaystyle \begin{aligned} R_a(u^+) = \lim_{\substack{\ \ v \to u^+\\ v \notin N_t}} R_a(v) \quad \text{and} \quad R_a(u^-) = \lim_{\substack{\ \ v \to u^-\\ v \notin N_t}} R_a(v), \end{aligned}$$

with one always turning to the right at a bifurcation point, the other always turning to the left. Every point of the little filled Julia set K _a is accumulated by at least one smooth or limit ray. When u ∈ N _t , only one of R _a (u ⁺) or R _a (u ⁻) can accumulate on K _a and we agree to denote this simply by R _a (u).

Consider the compact set

$$\displaystyle \begin{aligned} Y_t = \Big\{ u \in {\mathbb T} : m_3^{\circ i}(u) \notin \Big( t+\frac{1}{3}, t-\frac{1}{3} \Big) \ \text{for all} \ i \geq 0 \Big\}. \end{aligned}$$

It is not hard to show that Y _t contains a maximal m ₃-invariant Cantor set A _t characterized by the property that u ∈ A _t if and only if the (smooth or limit) ray R _a (u) accumulates on K _a . Every endpoint of a gap of A _t belongs to N _t and the inclusion $A_t \supset Y_t \smallsetminus N_t$ holds. According to [2], there exists a degree 1 monotone map $h:{\mathbb T} \to {\mathbb T}$, with plateaus over the gaps of A _t , which satisfies

$$\displaystyle \begin{aligned} h \circ m_3 = m_2 \circ h \quad \text{on} \quad A_t. \end{aligned} $$

(5.9)

The following is a special case of the main result of [26]:

Theorem 5.18

The ray R _a (u) with u ∈ A _t lands at z ∈ K _a if and only if the ray R(h(u)) of the quadratic P lands at ϕ _a (z) ∈ K(P).

Since K(P) is locally connected [23], it follows that all rays R _a (u) with u ∈ A _t land on K _a . In particular, since every point on the boundary of the Siegel disk of P is the landing point of one or two rays, and since $h|{ }_{A_t}$ is at most 2-to-1, we see that every point of ∂Δ _a is the landing point of at most four (smooth or limit) rays. An argument similar to Sect. 5.2 for connected Julia sets then shows that the set of angles of rays landing on ∂Δ _a contains a minimal rotation set X _a  ⊂ A _t under tripling, with ρ(X _a ) = θ. Let us investigate the relation between the deployment probability δ _a  ∈ [0, 1] of X _a and the external angle t of a.

We may assume without loss of generality that $s_0^+=\omega +\frac {1}{3} < t \leq t_0^+=\omega +\frac {5}{6}$ (the complementary case is treated by symmetry). Then the interval $(t+\frac {1}{3},t-\frac {1}{3})$ of length $\frac {1}{3}$ is contained in the major gap I ₀ of X _a that contains the fixed point 0. It will be convenient to first study the case where X _a  ∩ N _t  ≠ ∅, so at least one of the angles $t \pm \frac {1}{3}$ belongs to X _a . Since no angle in X _a is periodic under m ₃, the rays $R_a(t \pm \frac {1}{3})$ crash at e _a and then join as a single smooth path to land at a point w _a  ∈ ∂Δ _a which is characterized by the property that the internal angle from the non-escaping critical point c _a  ∈ ∂Δ _a to w _a is δ _a . Here are the possibilities:

• Case 1. δ _a  = 0. Then w _a  = c _a . We either have $I_0=(t,t-\frac {1}{3})$ where $t=\omega +\frac {5}{6}=t_0^+$, or $I_0=(t+\frac {1}{3},t)$ where $t=\omega +\frac {1}{2}=t_0^-$ (see Fig. 5.16a, b).

  Fig. 5.16

  

  Possible types of cubics f _a with $a \notin \mathbb {M}_3(\theta )$ which have a non-smooth ray landing on ∂Δ _a . (a) $\delta _a=0, t=t_0^+$. (b) $\delta _a=0, t=t_0^-$. (c) $\delta _a=n \theta , t=t_n^+$. (d) $\delta _a=n \theta , t=t_n^-$. (e) δ _a  = −nθ, t = t _n . (f) δ _a  ≠ nθ, t = t(δ _a )

• Case 2. $\delta _a=n\theta \ (\operatorname {mod} \ {\mathbb Z})$ for some n ≥ 1. Then $c_a=f_a^{\circ n}(w_a)$. We either have

  $$\displaystyle \begin{aligned} I_0=\Big( t+\frac{1}{3}-\frac{1}{3^{n+1}},t-\frac{1}{3} \Big), \quad \text{where} \quad t=\omega(\theta,n\theta)+\frac{1}{3}=t_n^+, \end{aligned}$$

  or

  $$\displaystyle \begin{aligned} I_0=\Big( t+\frac{1}{3},t-\frac{1}{3}+\frac{1}{3^{n+1}} \Big), \quad \text{where} \quad t=\omega(\theta,n\theta)+\frac{1}{3}-\frac{1}{3^{n+1}}=t_n^- \end{aligned}$$

  (see Fig. 5.16c, d which show the case n = 1).

• Case 3. $\delta _a=-n\theta \ (\operatorname {mod} \ {\mathbb Z})$ for some n ≥ 1. Then $w_a=f_a^{\circ n}(c_a)$ and we have $I_0=(t+\frac {1}{3},t-\frac {1}{3})$ where $t=\omega (\theta ,-n\theta )+\frac {1}{3}=t_n$ (see Fig. 5.16e which shows the case n = 1).

• Case 4. $\delta _a \neq n\theta \ (\operatorname {mod} \ {\mathbb Z})$ for all integers n. In this case c _a and w _a have disjoint orbits on ∂Δ _a , and we have $I_0=(t+\frac {1}{3},t-\frac {1}{3})$ where t = t(δ _a ) (see Fig. 5.16f).

Using monotonicity of δ↦ω(θ, δ), it is easy to see that the above cases classify X _a for all external angles t except when $t \in (t_n^-,t_n^+)$ for some n ≥ 0. As a corollary, we obtain

Corollary 5.19

If the external angle t of $a \notin \mathbb {M}_3(\theta )$ lies in $(t_n^-,t_n^+)$ for some n ≥ 0, then X _a is contained in the set

$$\displaystyle \begin{aligned} Y_t \smallsetminus N_t = \Big\{ u \in {\mathbb T} : m_3^{\circ i}(u) \notin \Big[ t+\frac{1}{3}, t-\frac{1}{3} \Big] \ \mathit{\mbox{for}\ \mbox{all}} \ i \geq 0 \Big\}. \end{aligned}$$

In particular, every dynamic ray R _a (u) with u ∈ X _a is smooth.

It remains to determine X _a when t belongs to such an interval. We will need a preliminary observation:

Lemma 5.20

Corollary 5.19 holds if we replace X _a with the rotation set X _θ,nθ.

Proof

We know that X _θ,nθ has a loose gap $I_0=(\alpha +\frac {1}{3}-\frac {1}{3^{n+1}},\alpha -\frac {1}{3})$ containing 0 and a taut gap $(\beta +\frac {1}{3},\beta -\frac {1}{3})$ containing $\frac {1}{2}$. Here

$$\displaystyle \begin{aligned} \alpha=\omega(\theta,n\theta)+\frac{1}{3} \quad \text{and} \quad \beta=\omega(\theta,-n\theta)-\frac{1}{6} \end{aligned}$$

(see Fig. 5.17). We have

$$\displaystyle \begin{aligned} t_n^- = \omega(\theta,n \theta)+\frac{1}{3}-\frac{1}{3^{n+1}} = \alpha-\frac{1}{3^{n+1}} \quad \text{and} \quad t_n^+ = \omega(\theta,n \theta)+\frac{1}{3} = \alpha, \end{aligned}$$

so the assumption $t_n^- < t < t_n^+$ implies $[ t+\frac {1}{3}, t-\frac {1}{3} ] \subset I_0$. Since the forward m ₃-orbit of every u ∈ X _θ,nθ avoids I ₀, it must avoid the subinterval $[ t+\frac {1}{3}, t-\frac {1}{3} ]$, which implies $u \in Y_t \smallsetminus N_t$. □

Fig. 5.17

Major gaps of X _θ,nθ and the proof of Lemma 5.20

Theorem 5.21

If the external angle t of $a \notin \mathbb {M}_3(\theta )$ lies in $(t_n^-,t_n^+)$ for some n ≥ 0, then X _a  = X _θ,nθ.

Proof

By Corollary 5.19, $X_a \subset Y_t \smallsetminus N_t \subset A_t$. The semiconjugacy h of (5.9) has plateaus over the gaps of A _t , so it is injective on X _a . Hence h maps X _a homeomorphically onto an m ₂-invariant Cantor set C = h(X _a ). If φ is the canonical semiconjugacy associated with X _a , the composition φ ∘ h ⁻¹ is a well-defined degree 1 monotone map of the circle since each fiber of h maps to a single point under φ. Since φ ∘ h ⁻¹ semiconjugates m ₂| _C to the rotation r _θ , it follows that C is a rotation set for m ₂ with ρ(C) = θ. Similarly, by Lemma 5.20 $X_{\theta ,n\theta } \subset Y_t \smallsetminus N_t \subset A_t$ and an identical argument shows that C′ = h(X _θ,nθ) is also a rotation set for m ₂ with ρ(C′) = θ. By the uniqueness of rotation sets under doubling, C = C′. It follows from injectivity of h that X _a  = X _θ,nθ. □

Assuming that the rays $\mathbb {R}(t_n^\pm )$ in fact land at a _n , we can define the parameter wake $\mathbb {W}_n$ as the connected component of ${\mathbb C} \smallsetminus (\mathbb {R}(t_n^-) \cup \mathbb {R}(t_n^+) \cup \{ a_n \})$ which does not meet Γ(θ). Using monotonicity of δ↦ω(θ, δ) it is easy to see that distinct parameter wakes are disjoint. Theorem 5.21 can be restated as saying that X _a  = X _θ,nθ whenever $a \in \mathbb {W}_n \smallsetminus \mathbb {M}_3(\theta )$. We can show that this holds for every $a \in \mathbb {W}_n$ (this contains the claim that X _a is well defined for $a \in \mathbb {W}_n \cap \mathbb {M}_3(\theta )$). The argument uses holomorphic motions as follows.

A dynamic ray R _a (u) moves holomorphically over the parameter $a \in {\mathbb C}$ as long as it remains smooth (see [6], Proposition 2). Lemma 5.20 shows that every ray R _a (u) with u ∈ X _θ,nθ is smooth for $a \in \mathbb {W}_n \smallsetminus \mathbb {M}_3(\theta )$. Since R _a (u) is trivially smooth for $a \in \mathbb {M}_3(\theta )$, it follows that this ray moves holomorphically over the entire parameter wake $\mathbb {W}_n$. By the λ-lemma, this motion extends to a holomorphic motion of the closure $\overline {R_a(u)}$ over $\mathbb {W}_n$. But for $a \in \mathbb {W}_n \smallsetminus \mathbb {M}_3(\theta )$ this closure is R _a (u) union its landing point on ∂Δ _a . Since ∂Δ _a also moves holomorphically over $\mathbb {W}_n$, it follows that R _a (u) lands on ∂Δ _a for every $a \in \mathbb {W}_n$, as required.

Away from the endpoints ± a ₀ of Γ(θ) the critical points of f _a can be continued analytically as a function of a (however, going around ± a ₀ will swap the two critical points, so the monodromy is non-trivial). In particular, the escaping and non-escaping critical points of f _a for $a \in \mathbb {W}_n \smallsetminus \mathbb {M}_3(\theta )$ extend to holomorphic functions a↦e _a , c _a defined for all $a \in \mathbb {W}_n$. The preceding paragraph then shows that e _a belongs to the dynamical wake $W(f_a^{-n}(c_a))$ whenever $a \in \mathbb {W}_n$. It seems likely that this property is the dynamical characterization of the parameter wake $\mathbb {W}_n$.

To summarize, we have identified the dependence of δ _a on a in the following cases:

• If $a \in \overline {\mathbb {W}_0}$, then δ _a  = 0.

• If $a \in \overline {-\mathbb {W}_0}$, then δ _a  = 1.

• If $a \in \overline {\mathbb {W}_n \cup \mathbb {R}(s_n)}$ for some n ≥ 1, then $\delta _a=n\theta \ (\operatorname {mod} \ {\mathbb Z})$.

• If $a \in \overline {-\mathbb {W}_n \cup \mathbb {R}(t_n)}$ for some n ≥ 1, then $\delta _a=-n\theta \ (\operatorname {mod} \ {\mathbb Z})$.

• If $a \in \overline {\mathbb {R}(t(\delta )) \cup \mathbb {R}(s(\delta ))}$ where $\delta \neq n \theta \ (\operatorname {mod} \ {\mathbb Z})$ for all n, then δ _a  = δ.

It is conjectured that an analog of the limb decomposition hypothesis in Sect. 5.1 holds in this cubic parameter space, in the sense that the parameter limbs $\mathbb {L}_n = \mathbb {M}_3(\theta ) \cap \overline {\mathbb {W}_n}$ have shrinking diameters as n →∞. Under this assumption, the connectedness locus $\mathbb {M}_3(\theta )$ would be the union of the arc Γ(θ) together with the parameter limbs $\pm \mathbb {L}_n$ for all n ≥ 0, and the five cases above would describe δ _a (hence X _a ) for every $a \in {\mathbb C}$.