Homeomorphisms between Limbs of the Mandelbrot Set

Bodil Branner The Tech. Univ. of Denmark Building 303 DK-2800 Lyngby, Denmark

         brannermat.dtu.dk

Núria Fagella Math. Sci. Research Inst. 1000 Centennial Drive Berkeley, CA 94720, USA

                                    nuriamath.bu.edu

Abstract

Using a family of higher degree polynomials as a bridge, together with complex surgery techniques, we construct a homeomorphism between any two limbs of the Mandelbrot set of common denominator. Induced by these homeomorphisms and complex conjugation we obtain an involution between each limb and itself, whose fixed points form a topological arc. All these maps have counterparts at the combinatorial level relating corresponding external arguments. Assuming local connectivity of the Mandelbrot set we may conclude that the constructed homeomorphisms between limbs are compatible with the embeddings of the limbs in the plane. As usual we plough in the dynamical planes and harvest in the parameter space.

Contents

1 Introduction 3

2 Preliminaries 11

2.1 Dynamics of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 The Quadratic family . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Families of Higher Degree Polynomials 16 3.1 Monic Families of Higher Degree Polynomials and External Rays . . . . . . 21 3.2 External Rays in the 0-wake . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Dynamical Characterizations 25

4.1 Q_{c} with c in a rational wake . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.1 Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 P_{q,λ} with λ in the 0-wake . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Surgery: Proof of Theorem F 31 5.1 Idea of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Construction of a polynomial of degree q + 1. Definition of ϕ_{p/q}. . . . . . . 32 5.3 Bijectivity of ϕ_{p/q} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.4 Analyticity of ϕ_{p/q} in the interior of M_{p/q} . . . . . . . . . . . . . . . . . . . 51 5.5 Continuity of ϕ_{p/q} at points on the boundary of M_{p/q} . . . . . . . . . . . . . 56 5.6 Proof of theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.7 Compatibility with Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Involutions: Proof of Theorem B 61

7 Combinatorial Surgery:

Proof of Theorems H, C, D and E 63 7.1 Proof of Theorems H and C . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.1.1 Dynamical Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.1.2 Parameter Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.2 Proof of Theorem D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.3 Proof of Theorem E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

1 Introduction

The Mandelbrot set has been a main object of interest and study in the recent years. It is associated to the quadratic family of complex polynomials Q_{c}(z) = z^{2} + c, for c ∈ ℂ. Let K_{c} denote the filled Julia set of Q_{c}, i.e.

K = K_{c} = {z ∈ ℂ | {Q_{c}^{n}(z)}_{n∈ℕ} is bounded },

and let J = J_{c} be the boundary of K_{c}, the Julia set of Q_{c}.

As for all rational functions, the dynamical behavior of the critical point ω = 0 dominates the dynamical behavior of the polynomial. The filled Julia set K_{c} is connected when the orbit of 0 is bounded and totally disconnected when it is unbounded.

The Mandelbrot set, M, is defined as the set of parameter values c, for which K_{c} is connected, or equivalently as the set of parameters for which the orbit of 0 is bounded.

The works of A. Douady, J. H. Hubbard [DH1] [DH2], D. Sullivan [MSS] and J-C. Yoccoz in the last decade contributed enormously to the understanding of the Mandelbrot set, but there remain many interesting open questions. The main one is to prove that M is locally connected (MLC conjecture). The following are some well known results about M:

1. M is full, compact and connected.

2. The interior of M contains connected components for which Q_{c} has an attracting periodic orbit. These are called hyperbolic components and it is conjectured that their union equals the interior of M.

3. For each hyperbolic component Ω, there is a conformal isomorphism ρ_{Ω} : 𝔻 → Ω such that Q_{c} has a cycle of multiplier t when c = ρ_{Ω}(t). The point ρ_{Ω}(0) is called the center of Ω. This map extends to a homeomorphism of \overline{𝔻} onto \overline{Ω}. The function γ(t) = γ_{Ω}(t) = lim_{r→1} ρ_{Ω}(re^{2πit}) defines a parametrization of the boundary of Ω. For each t ∈ ℝ/ℤ, the point γ(t) in ∂Ω is said to have internal argument t. The point γ(0) is called the root of the hyperbolic component Ω.

4. There is a unique hyperbolic component, namely Ω_{0}, that is bounded by the main cardioid, for which Q_{c} has an attracting fixed point . For any internal argument p/q ∈ ℚ/ℤ, there is a hyperbolic component Ω_{p/q} attached to the cardioid at the point γ_{0}(p/q). This hyperbolic component contains c-values for which Q_{c} has an attracting periodic cycle of period q. The point γ_{0}(p/q) is the root of the hyperbolic component Ω_{p/q}.

5. There exists a conformal isomorphism φ_{M} : ℂ \ M ⟶ ℂ \ \overline{𝔻} such that \frac{φ_{M} (c)}{c} → 1 when c → ∞. An external ray of external argument θ is defined by

R_{M}(θ) = φ_{M}^{−1}({re^{2πiθ}}_{1<r<∞}).

(See Figure 1 and Sect. 2.1.1). Rays with rational arguments play a special role. They are known to have a limit on the boundary of M as r → 1. The set of landing points of these rays consists of all the roots of the hyperbolic components and all the Misiurewicz points, i.e. c values for which ω = 0 is strictly preperiodic.

Figure 1: The Mandelbrot set.

Let p and q be positive integers, p < q, q ≥ 2 and gcd(p, q) = 1.

We define the p/q-limb of M, M_{p/q}, to be the connected component of M \ \overline{Ω}_{0} attached to the main cardioid at the point with internal argument p/q, i.e. at c = γ_{0}(p/q) (see Figure 1).

We define the p/q-wake of M, W M_{p/q}, to be the open subset of ℂ that contains the p/q-limb of M and is bounded by the two rays landing at c = γ_{0}(p/q) union this point [A].

Definition Given two compact sets K and K′ in ℂ, we say that a homeomorphism ϕ : K → K′ is compatible (resp. reversely compatible) with the embeddings of K and K′ in ℂ if there are neighborhoods N and N′ of K and K′ respectively such that ϕ extends to a homeomorphism ϕ̂: N → N′, preserving (resp. reversing) orientation.

Some years ago, J. C. Yoccoz observed (unpublished) that the limbs M_{1/5} and M_{2/5} are homeomorphic and that corresponding polynomials have conjugated dynamics on the filled Julia sets. Recently E. Lau and D. Schleicher [LS] have announced the existence of homeomorphisms between any two limbs with common denominator compatible with the dynamics, but not compatible with the embeddings of the limbs in the plane.

In this paper, we construct a homeomorphism between any two limbs with common denominator. The homeomorphism is not compatible with the dynamics but compatible with the embeddings assuming local connectivity of M. In fact, we believe that the hypothesis of local connectivity could be removed but we will discuss this fact in a forthcoming paper.

Our goal is to prove the following theorems:

Theorem A Given p/q and p′/q, there exists a homeomorphism

Φ = Φ_{pp′}^{q} : M_{p/q} ⟶ M_{p′/q}

which is holomorphic on the interior of M_{p/q}.

Moreover, for each c ∈ M_{p/q}, there exists a homeomorphism

Φ̂_{c} : K_{c} ⟶ K_{Φ(c)}

which is holomorphic in the interior of K_{c} and compatible with the embeddings of K_{c} and K_{Φ(c)} in ℂ.

See Figs. 2, 3 and 4. In fact, a stronger statement is true: for all c ∈ M_{p/q} the homeomorphism Φ̂_{c} matches with the combinatorially defined map described in Theorem C.

Theorem B Given p/q, there exists a homeomorphism,

I = I_{p/q} : M_{p/q} ⟶ M_{p/q}.

which is an involution, i.e. I^{2} = Id, and is anti-holomorphic in the interior of M_{p/q}. The set of points that are fixed by I form a topological arc through the limb.

Moreover, for each c ∈ M_{p/q}, there exists a homeomorphism

Î = Î_{c} : K_{c} ⟶ K_{I(c)}

reversely compatible with the embeddings, which is anti-holomorphic in the interior of K_{c}.

See Figs. 5, 6 and 7.

Figure 2: The 1/5-limb and the 2/5-limb of the Mandelbrot set.

Figure 3: Corresponding filled Julia sets under ϕ̂_{c}. Left: c ∈ M_{1/5}, center of a period 6 hyperbolic component. Right: c ∈ M_{2/5}, center of a period 8 hyperbolic component.

Figure 4: Hubbard trees for the filled Julia sets in Fig. 3 Figure 5: Arc of symmetry for the involutions I_{1/3} and I_{1/4}.

Figure 6: Arc of symmetry for the involution I_{1/19}.

Figure 7: The Julia sets for the landing point of the ray R_{M}(1/4) and I_{1/3}(c) = i.

Theorem C

Given p/q and p′/q, let θ_{p/q}^{±} and θ_{p′/q}^{±} be the arguments of the external rays landing at the root points of the limbs M_{p/q} and M_{p′/q} respectively. Then, there exists an orientation preserving homeomorphism

Θ_{pp′}^{q} : [θ_{p/q}^{−}, θ_{p/q}^{+}] ⟶ [θ_{p′/q}^{−}, θ_{p′/q}^{+}]

such that

1. For θ ∈ [θ_{p/q}^{−}, θ_{p/q}^{+}] ∩ ℚ, the ray R_{M}(θ) lands at a point c ∈ M_{p/q} if and only if R_{M}(Θ_{pp′}^{q}(θ)) lands at Φ_{pp′}^{q}(c) ∈ M_{p′/q}.

2. The map Θ_{pp′}^{q} is induced by a homeomorphism

Θ̂ = Θ̂_{pp′}^{q} : 𝕋 ⟶ 𝕋

such that for any θ ∈ 𝕋 and any c ∈ M_{p/q}, the ray R_{c}(θ) lands at z ∈ K_{c} if and only if the ray R_{Φ_{pp′}^{q}(c)}(Θ̂(θ)) lands at Φ̂_{c}(z) ∈ K_{Φ_{pp′}^{q}(c)}.

It is remarkable that the map Θ_{pp′}^{q} in Theorem C does not depend on the point c ∈ M_{p/q}. A stronger statement than part one of Theorem C is in fact true (see remark 7.1.2).

Theorem D

Given p/q and θ_{p/q}^{±} as in Theorem C, there exists an orientation reversing homeomorphism \overline{Θ} = \overline{Θ}_{p/q} of [θ_{p/q}^{−}, θ_{p/q}^{+}] onto itself such that

1. \overline{Θ}^{2} = Id.

2. For θ ∈ [θ_{p/q}^{−}, θ_{p/q}^{+}] ∩ ℚ, the ray R_{M}(θ) lands at a point c ∈ M_{p/q} if and only if R_{M}(\overline{Θ}(θ)) lands at I_{p/q}(c) ∈ M_{p/q},

3. The argument θ^{s} fixed by \overline{Θ} is rational.

4. The map \overline{Θ}_{p/q} is induced by an orientation reversing homeomorphism

ˆ\overline{Θ} = ˆ \overline{Θ}_{p/q} : 𝕋 ⟶ 𝕋

such that given c ∈ M_{p/q} and θ ∈ 𝕋, the ray R_{c}(θ) lands at a point z ∈ K_{c}, if and

only if the ray R_{I(c)}( ˆ\overline{Θ}(θ)) lands at Î(z) ∈ K_{I(c)}.

Figure 8: Some rays with corresponding arguments under the map \overline{Θ}_{1/3}.

Theorem E

Assume M is locally connected. Then the map Φ_{pp′}^{q} (resp. I_{p/q}) is compatible (resp. reversely compatible) with the embeddings of the limbs in ℂ.

To prove Theorem A, we use complex surgery techniques which are analogous to those used in [BD], where a homeomorphism was constructed between the 1/2-limb of M and a limb in a cubic parameter space. The way we use this technique is somewhat unexpected, since we use different families of higher degree polynomials, P_{q,λ} = λz(1 + z/q)^{q}, as a bridge for the construction of the homeomorphisms. That is, we actually show that each of the p/q-limbs of M is homeomorphic to the 0-limb L_{q,0} to be defined in section 3. These families of polynomials were introduced in [F] as a way to approximate the family z ↦ λze^{z}, inspired by [DGH] where they considered polynomial families approximating z ↦ λe^{z}.

The homeomorphisms could be defined directly without reference to the higher degree polynomials. But the similarity between corresponding quadratic polynomials becomes more transparent when passing to these special higher degree polynomials. Moreover the involutions together with the arc of symmetry on the limbs are immediate from the symmetry of the 0-limb L_{q,0} with respect to the real axis. Although the involutions also follow from the symmetry of M with respect to the real axis together with the homeomorphisms in Theorem A, it is not clear that the fixed points form a topological arc.

The statements given above for theorems A to D just mention existence of homeomorphisms. In this paper we actually construct such homeomorphisms. The homeomorphisms constructed are not compatible with the dynamics.

The paper is structured as follows:

Instead of proving Theorem A-D directly we prove analogous theorems F-H, forming the bridges to higher degree polynomials. In Sect. 2 we recall some general results which we use throughout the paper. In Sect. 3 we introduce the families of higher degree polynomials that we use as bridges, and formulate Theorem F, the parameter space analogue of Theorem A. In Sect. 4 we emphasize the common dynamical features to be applied. Sect. 5 is by far the most important part of the paper. Using surgery techniques we prove Theorem F and also Theorem G, the dynamical analogue of Theorem A. Theorem C follows immediately. Sect. 6 is dedicated to Theorem B. In the last section we state and prove Theorem H, the combinatorial analogue of Theorem C. Theorem C-E follow.

We thank Bob Devaney, Alfredo Poirier, Dan Sørensen, Dierk Schleicher and Tan Lei for many helpful conversations and especially, we thank Adrien Douady not only for innumerable comments and suggestions but also for his constant encouragement. We also thank Christian Mannes for creating the program that produced all the pictures in this paper. We wish to thank the Mathematical Sciences Research Institute at Berkeley, the Mathematics Department at Boston University and the Mathematical Institute at the Technichal University of Denmark for their hospitality. Research at MSRI was supported in part by NSF grant DMS-9022140.

2 Preliminaries

2.1 Dynamics of Polynomials

If f is a complex polynomial of degree d > 1, the point at infinity is always a superattracting fixed point. We define its filled Julia set as

K_{f} = {z ∈ ℂ | {f^{n}(z)}_{n∈ℕ} is bounded},

and its Julia set J_{f} as the boundary of K_{f}.

Proposition 2.1 Let f be a polynomial of degree d. Then, K_{f} is connected if and only if the orbits of all critical points of f are bounded.

For a proof see [Bl]

Assume f is monic and K_{f} connected. Then, there exists a unique analytic isomorphism tangent to the identity at infinity, ψ_{f} : ℂ \ \overline{𝔻} ⟶ ℂ \ K_{f}, which conjugates f_{0}(z) = z^{d} to f. The map ψ_{f} is called the Böttcher coordinate.

We can also lift f_{0} to a map f̃_{0} on the universal covering space, the right half plane ℍ. If we require that f̃_{0}(ℝ_{+}) = ℝ_{+} we have that

f̃_{0}(z) = M_{d}(z) := dz

In summary, the following diagram commutes

ℍ \underrightarrow{_{f_{0}}} ℍ
exp↓  ↓^{exp}
ℂ \ \overline{𝔻} ⟶^{f_{0}} ℂ \ \overline{𝔻}
ψ_{f} ↓  ↓^{ψ_{f}}
ℂ \ K_{f} ⟶^{f} ℂ \ K_{f}

We remark that in the case of K_{f} being locally connected, ψ_{f} extends continuously to the boundary of 𝔻, so that ψ_{f} ∘ exp is defined on \overline{ℍ}. Even in the case when K_{f} is not locally connected, there is a set of points of full measure on ∂𝔻 where ψ_{f} is well defined. This set always includes the points with rational arguments.

Given t ∈ ℝ we denote by R(t) the horizontal line in ℍ with imaginary part equal to 2πt i.e.

R(t) := {ρ + 2πit ∈ ℍ | ρ > 0}.

We define the external ray of argument t to be

R_{f}(t) = ψ_{f}(exp(R(t)))
= ψ_{f}({(re^{2πit}) | r ∈ (1, ∞)}).

If R_{f}(t) has a limit when r → 1 (or ρ → 0), then it tends to a point of the Julia set which we denote by R_{f}^{∗}(t). We say that R_{f}(t) lands at R_{f}^{∗}(t) and we have:

f(R_{f}^{∗}(t)) = R_{f}^{∗}(dt).

All external rays with rational arguments land and if K_{f} is locally connected all external rays land.

The potential, G_{f}, (Green’s function) of K_{f} satisfies G_{f} = log(|ψ_{f}^{−1}|) on ℂ \ K_{f}. We may extend it to ℂ defining G_{f} = 0 on K_{f}. We have that

G_{f}(f(z)) = d G_{f}(z)

for all z ∈ ℂ. The positive level sets of the potential function are simple closed curves around K_{f} called equipotentials, and the potential measures the rate of escape of points under iteration. See Figure 9. A given equipotential {z ∈ ℂ \ K_{f} | G_{f}(z) = η} corresponds in the exterior of the unit disc to a circle around the origin of radius e^{η} and on the right half plane to a vertical line of real part η.

Figure 9: Sketch of an external ray and some level sets of the potential function.

2.1.1 The Quadratic family

Let Q_{c}(z) = z^{2} + c and set K_{c} = K_{Q_{c}}, ψ_{c} = ψ_{Q_{c}}, etc. Let M be the Mandelbrot set (see Sect. 1). The results in this section can be found in [DH2].

The map φ_{M} : ℂ \ M ⟶ ℂ \ \overline{D} defined as φ_{M}(c) = ψ_{c}^{−1}(c) is a conformal isomorphism. So we define an external ray of external argument θ as

R_{M}(θ) = φ_{M}^{−1}({re^{2πiθ}}_{1<r<∞}).

If R_{M}(θ) has a limit c ∈ ∂M when r → 1 we say that R_{M}(θ) lands at c. We denote the landing point of the ray R_{M}(θ) by R_{M}^{∗}(θ).

The conjecture that the Mandelbrot set is locally connected is equivalent to the continuous landing of all external rays.

It is known that all external rays with rational arguments land. In what follows we relate the external rays of M with the external rays in the dynamical plane of the corresponding quadratic polynomials, as well as the relation between the landing points in the case of rational arguments.

1. Suppose c ∈ ℂ \ M. Then

c ∈ R_{M}(θ) ⟺ c ∈ R_{c}(θ)  (1)

2. Suppose c is a Misiurewicz point in M. Then

R_{c}(θ) lands at c ∈ K_{c} ⟺ R_{M}(θ) lands at c ∈ M.

An argument θ satisfying the above is rational with even denominator.

3. Suppose c is the center of a hyperbolic component Ω of M of period k. Let c_{0} = γ_{Ω}(0) denote the root point of the hyperbolic component. Let U_{c} denote the connected component of Int(K_{c}) containing c and let z_{0} denote the root point on ∂U_{c}, that is the boundary point which is fixed by Q_{c}^{k}. Then

R_{c}(θ) lands at z_{0} ∈ K_{c} adjacent to U_{c} ⟺ R_{M}(θ) lands at c_{0} ∈ M.

An argument θ satisfying the above is rational with odd denominator.

2.2 Tools

For the surgery in Sect. 5 we will use the theory of quasi-conformal mappings, the theory of integrability by Morrey-Ahlfors-Bers and the theory of Polynomial-like mappings of Douady and Hubbard (compare with [BD], [DH3]).

Quasi-conformal mappings, Beltrami forms and almost complex structures

Let X and Y be two Riemann surfaces isomorphic either to 𝔻 (the unit disc) or ℂ.

An almost complex structure σ on X is a measurable field of ellipses (E_{x})_{x∈X}, equivalently defined by a measurable Beltrami form μ on X

μ = u\frac{d\overline{z}}{dz} .

The correspondence between Beltrami forms and complex structures is as follows: the argument of u(x) is twice the argument of the major axis of E_{x}, and |u(x)| = \frac{K−1}{K+1} where K ≥ 1 is the ratio of the lengths of the axes.

The standard complex structure σ_{0} is defined by circles.

Suppose φ : X ⟶ Y is a quasi-conformal homeomorphism. Then φ gives rise to an almost complex structure σ on X. For almost every x ∈ X, φ is differentiable and the ℝ-linear tangent map T_{x}φ : T_{x}X ⟶ T_{φ(x)}Y defines, up to multiplication by a positive factor, an ellipse E_{x} in T_{x}X:

E_{x} = (T_{x}φ)^{−1}(S^{1}).

Moreover, there exists a constant K > 1 such that the ratio of the axes of E_{x} is bounded by K for almost every x ∈ X. The smallest bound is called the dilatation ratio of φ.

Equivalently, φ defines a measurable Beltrami form on X

μ = \frac{\overline{∂}φ}{∂φ} = \frac{\frac{∂φ}{∂\overline{z}}}{\frac{∂φ}{∂z}} \frac{d\overline{z}}{dz} = u(z)\frac{d\overline{z}}{dz} .

An almost complex structure is quasi-conformally equivalent to the standard structure if it is defined by a measurable field of ellipses with bounded dilatation ratio.

Given φ : X ⟶ Y a quasi-conformal homeomorphism, an almost complex structure σ on Y can be pulled back into an almost complex structure φ^{∗}σ on X. If σ is defined by an infinitesimal field of ellipses (E_{y})_{y∈Y} then φ^{∗}σ is defined by (E_{x})_{x∈X} where E_{x} = (T_{x}φ)^{−1}E_{φ(x)} whenever defined.

To integrate an almost complex structure σ means to find a quasi-conformal homeomorphism φ such that (T_{x}φ)^{−1}(S^{1}) = ρ(x)E_{x} for almost every x ∈ X. Informally, we will say that σ is transported to σ_{0} by φ.

A quasi-regular mapping is of the form ψ = h ∘ φ where φ is quasi-conformal and h is holomorphic, but h may have critical points.

The Theorem of Integrability can be stated as follows:

Theorem of Integrability (Measurable Riemann Mapping Theorem)

Let X be a Riemann surface isomorphic to 𝔻 or to ℂ. Let σ_{μ} be any almost complex structure on X given by the Beltrami form

μ = u\frac{d\overline{z}}{dz}

with bounded dilatation ratio, i.e.

∥μ∥_{∞} := sup|u(z)| < m < 1.

Then σ_{μ} is integrable i.e. there exists a quasi-conformal homeomorphism φ such that

μ = \frac{\overline{∂}φ}{∂φ}.

If X is isomorphic to 𝔻 then φ : X ⟶ 𝔻 is unique up to composition with a Möbius transformation mapping 𝔻 to 𝔻. If X is isomorphic to ℂ then φ : X ⟶ ℂ is unique up to composition with an affine map.

In Sect. 5.5 we will need the dependence on parameters of the integrating maps, when dealing with Beltrami forms defined in ℂ.

Theorem of Integrability (Dependence on Parameters)

Let Λ be an open set of ℂ and let (μ_{λ} = u_{λ} \frac{d\overline{z}}{dz})_{λ∈Λ} be a family of measurable Beltrami forms on ℂ. Suppose that λ ↦ u_{λ}(z) is holomorphic for each fixed z ∈ ℂ, and that there is m < 1 such that ∥μ_{λ}∥_{∞} < m for all λ ∈ Λ. Let φ_{λ} : ℂ → ℂ be the unique quasi-conformal homeomorphisms that fixes two given points in ℂ and such that μ_{λ} = \frac{\overline{∂}φ_{λ}}{∂φ_{λ}}. Then for each z ∈ ℂ the map λ ↦ φ_{λ}(z) is holomorphic.

Polynomial-like mappings

A polynomial-like mapping is a proper holomorphic mapping

f : U′ ⟶ U

where U′ and U are open sets in a Riemann surface X isomorphic to a disc, and \overline{U′} ⊂ U. The degree of f is the number of preimages of any point z ∈ U counted with multiplicity.

For a polynomial-like mapping f we define

K_{f} = {z ∈ U′ | f^{n}(z) ∈ U′ for all n ∈ ℕ}.

Given two polynomial-like mappings f : U′ → U and g : V′ → V with K_{f} and K_{g} connected we say that f and g are holomorphically equivalent (resp. quasi-conformally equivalent) if there exists a holomorphic (resp. quasi-conformal) homeomorphism

φ : U_{1} ⟶ V_{1}

where U_{1} and V_{1} are neighborhoods of K_{f} and K_{g} conjugating f to g on U′_{1} = f^{−1}(U_{1}) i.e. satisfying g ∘ φ = φ ∘ f on U′_{1}.

A hybrid equivalence is a quasi-conformal equivalence which satisfies \overline{∂}φ = 0 almost everywhere on K_{f} so which is holomorphic in the interior of K_{f}.

If f and g are polynomials, a co-hybrid equivalence between f and g is a quasi-conformal equivalence φ : ℂ → ℂ which is holomorphic on ℂ \ K_{f}.

The Straightening Theorem Let f : U′ ⟶ U be a polynomial-like mapping of degree d with K_{f} connected. Then, f is hybrid equivalent to a polynomial P of degree d. Moreover, this polynomial is unique up to an affine conjugacy.

We will also need the following result:

Proposition 2.2 Let f and g be polynomials with K_{f} and K_{g} connected. If f and g are hybrid equivalent then they are conjugate by an affine map.

3 Families of Higher Degree Polynomials

Consider the family of complex polynomials P_{q,λ}(z) = λz(1 + z/q)^{q}, where λ ∈ ℂ \ {0} and q ≥ 2. Each P_{q,λ} has degree q + 1 and has exactly two critical points: −q with multiplicity q − 1, and ω = \frac{−q}{q+1} with multiplicity one. Moreover, P_{q,λ}(−q) = 0 and 0 is fixed. We now show that polynomials with these features, can always be expressed in the form P_{q,λ}(z).

Definition A polynomial P (z) of degree q + 1 ≥ 3 is of type E_{q} if it satisfies the following properties:

1. P has two critical points: −q of multiplicity q − 1 and ω ∈ ℂ of multiplicity one.

2. P has a fixed point at z = 0.

3. P (−q) = 0.

Proposition 3.1 Any polynomial P (z) of degree q + 1 which is of type E_{q} is of the form

P_{q,λ}(z) = λz(1 + \frac{z}{q})^{q}.

Moreover, if P_{q,λ} and P_{q,λ′} of type E_{q} are affine conjugate with q ≥ 3, then λ = λ′.

From here and throughout the paper, we will omit the subindex q whenever it causes no confusion. Proof : The polynomial P has a zero of order q at the point −q and a simple zero at 0 and, since P is of degree q + 1, it is of the form λz(1 + z/q)^{q}.

For q ≥ 3, if an affine map A conjugates P_{q,λ} to P_{q,λ′} it must fix −q (because it is the only multiple critical point for both polynomials) and 0 (because it is the image of −q), so A is the identity.

q.e.d.

Since the critical point of high multiplicity is always prefixed, the dynamics of P_{q,λ} are dominated by the behavior of the free critical point, ω. We have the following lemma, which follows from Prop. 2.1:

Lemma 3.2 Let K_{q,λ} be the filled Julia set of P_{q,λ}. Then, K_{q,λ} is connected if and only if the orbit of ω is bounded.

Proposition 3.3 Let P_{q,λ} and P_{q,λ′} be polynomials of type E_{q} with K_{λ} and K_{λ′} connected. If they are hybrid equivalent, then λ = λ′.

Proof : This is a direct corollary of Prop. 2.2. By this proposition P_{q,λ} and P_{q,λ′} are affine conjugate. By the Prop. 3.1 λ = λ′.

q.e.d.

For each q ∈ ℕ let Λ_{q} be ℂ^{∗} viewed as the parameter space of P_{q,λ}. Let L_{q} denote the connectedness locus, i.e. the set of λ ∈ Λ_{q} for which K_{q,λ} is connected. Observe that the set L_{q} associated with P_{q,λ} is the analogue of the Mandelbrot set for Q_{c}. Figures 10, through 15 show the sets L_{1}, L_{2}, L_{3}, L_{4}, L_{5} and L_{19}, respectively. Following this analogy, we will call a connected component Ω of the interior of L_{q} a hyperbolic component if the free critical point ω is attracted to an attracting cycle (despite the fact that the polynomial is not hyperbolic, since the orbit {−q, 0} is contained in the Julia set, see Sect. 3.1). As in the Mandelbrot set, a hyperbolic component Ω is parametrized by the multiplier map ρ_{Ω} : 𝔻 ⟶ Ω, a biholomorphic map that extends continuously to the boundary. For all q, the sets L_{q} share the following features:

1. If 0 < |λ| < 1, z = 0 is an attracting fixed point of P_{q,λ}, with ω in its basin of attraction. Hence the punctured closed unit disk is contained in L_{q}.

2. Let r/s ∈ ℚ/ℤ such that r ≥ 0, s > 0 and gcd(r, s) = 1. At λ_{\frac{r}{s}} = e^{2πi \frac{r}{s}}, the polynomial P_{q,λ} experiences a period s-tupling bifurcation. That is, when we exit the unit disk through λ_{\frac{r}{s}}, z = 0 becomes repelling and a cycle of period s becomes attracting. Hence, there is a hyperbolic component of period s attached to the unit disk at λ_{\frac{r}{s}}.

Figure 10: Parameter plane Λ_{1}: the Connectedness Locus L_{1} and the 0-wake. Range: [−2.5, 9.14]× [−2.1, 2.1].

For L_{q}, the unit circle is the analogue to the main cardioid of M. Here, we do not have a cusp and a hyperbolic component of period 1 is attached to λ_{\frac{0}{1}} = 1.

As we did for M, we define the \frac{r}{s}-limb of L_{q}, L_{q, \frac{r}{s}}, to be the connected component of L_{q} \ \overline{𝔻} attached to the unit circle at the point λ_{\frac{r}{s}}. The 0-limb, L_{q,0}, will be of special importance in our work.

One can define external rays in ℂ \ L_{q} similarly to ℂ \ M, to be explained in section 3.2. There are exactly two rational rays landing at λ = 1, the root of 0-limb. Hence we may define the 0-wake W L_{q,0} of L_{q}, as the open subset of ℂ that contains the 0-limb of L_{q} and is bounded by the two rays landing at its root. In section 4.2, the 0-wake is discussed in detail.

Theorem A is now a corollary of the following:

Figure 11: Parameter plane Λ_{2}: the Connectedness Locus L_{2} and the o-wake. Range: [−2.2, 4.4] × [−3.5, 3.5].

Figure 12: Parameter plane Λ_{3}: the Connectedness Locus L_{3} and the 0-wake. Range: [−3, 13] × [−4.7, 4.7].

Figure 13: Parameter plane Λ_{4}: the Connectedness Locus L_{4} and the 0-wake. Range: [−3.5, 23.4]× [−9.5, 9.5].

Figure 14: Parameter plane Λ_{5}: the Connectedness Locus L_{5}andthe0 − wake.

Figure 15: Parameter plane Λ_{19}: the Connectedness Locus L_{19} and the 0−wake. Range: [−6, 95]× [−45.6, 45.6].

Theorem F

Let p and q be positive integers such that p < q, q ≥ 3 and gcd(p, q) = 1. Then, there exists a homeomorphism

ϕ = ϕ_{p/q} : M_{p/q} ⟶ L_{q,0},

which is holomorphic in the interior of M_{p/q}.

Note that we have restricted to q ≥ 3. The reason is that the connectedness locus L_{1} is a ramified double covering of M and not related to limbs of M; the 0-limb L_{2,0} of L_{2} is homeomorphic to M_{1/2} [BD] and gives no new information with respect to Theorems A to E.

3.1 Monic Families of Higher Degree Polynomials and External Rays

Our goal in this section is to make precise definitions of external rays in the dynamical plane of P_{q,λ} as well as in the parameter plane. Also, we characterize the polynomials P_{q,λ} with λ in the 0-wake.

As explained in Sect. 2.1, when P is a monic polynomial, the ray R_{P} (θ) is define using the Böttcher coordinates ψ_{P} which is tangent to the identity at infinity. Here we are dealing with P_{q,λ} which is not monic, hence there is no canonical choice of a map conjugating P_{q,λ} to z ↦ z^{q+1} on the complement of the filled Julia set. This will cause some minor complication.

Set

P̃_{ν}(z) = P̃_{q,ν}(z) = z(z + ν)^{q}.

The polynomial P̃_{q,ν} is affine conjugate to P_{q,λ} with λ = ν^{q} through the map g_{ν}(z) = \frac{q}{ν} z. Under this map, the special fixed point 0 is still at 0, the multiple critical point is at −ν, the free critical point is at ω̃_{ν} = \frac{−ν}{q+1} and the free critical value is at

ṽ_{ν} = P̃_{ν}(\frac{−ν}{q + 1}) = −\frac{q^{q}}{(q + 1)^{q+1}} ν^{q+1}.  (2)

Let K̃_{ν} = K_{P_{ν}} and ψ̃_{ν} : ℂ \ \overline{𝔻} ⟶ ℂ \ K̃_{ν} denote the Böttcher coordinates for P̃_{ν}, i.e. conjugating P̃_{ν} to z ↦ z^{q+1} and tangent to the identity at infinity.

For λ ∈ L_{q} \ ℝ_{−} let ψ_{λ} : ℂ \ \overline{𝔻} ⟶ ℂ \ K_{λ} denote the unique Böttcher coordinate conjugating P_{λ} to z ↦ z^{q+1} and being asymptotic to z ↦ \frac{q}{λ^{1/q}} z at ∞, where ()^{1/q} denotes the principal branch of the q-th root. In other words ψ_{λ} = g_{ν} ∘ ψ̃_{ν} with ν = λ^{1/q}.

We may now define the external ray of argument t to be

R_{λ}(t) = ψ_{λ}(exp(R(t)))
= ψ_{λ}({re^{2πit} | r ∈ (1, ∞)})

and denote G_{λ} = log |ψ_{λ}^{−1}| the potential function.

Proposition 3.4 A parameter λ belongs to the 0-wake W L_{q,0} if and only if z = 0 is a repelling fixed point of P_{q,λ} and the landing point of a fixed ray.

In order to show this proposition we will need to define the external rays in parameter space. We will work again with the monic polynomials.

Let L̃_{q} denote the connectedness locus of the monic family (see Fig. 16), i.e.

L̃_{q} = {ν ∈ ℂ | K̃_{ν} connected}.

Set Λ̃_{q} = ℂ^{∗} and let Π = Π_{q} : Λ̃_{q} ⟶ Λ_{q} denote the q-fold covering map defined by Π(ν) = ν^{q}. Note that Π maps the wedge Σ_{q} = \{ν ∈ Λ̃_{q} | |Arg(ν)| < \frac{π}{2q} \} conformally onto Λ_{q} \ ℝ_{−} and in particular, the 0-limb L̃_{q,0} onto the 0-limb L_{q,0}.

Let ψ_{L_{q}} : ℂ \ L̃_{q} ⟶ ℂ \ \overline{𝔻} denote the unique Riemann mapping asymptotic to ν ↦ k̃ν

at ∞, where k̃ is some positive real number. The relation between the dynamical maps and the parameter maps is contained in the next proposition.

Figure 16: The connectedness loci L̃_{3} and L_{3}.

Proposition 3.5 Let ν ∈ ℂ \ L̃_{q}. Then

ψ̃_{ν}^{−1}(ṽ_{ν}) = −(ψ_{L_{q}}(ν))^{q+1}.  (3)

Proof : Note that the map

\begin{matrix}ℂ \ L̃_{q} & ⟶ & ℂ \ \overline{𝔻} \\ ν & ⟼ & ψ̃_{ν}^{−1}(ṽ_{ν}) \end{matrix}

is a (q + 1)-covering map and that the asymptotic behavior as ν → ∞ is given by eq. (2). Moreover the map

\begin{matrix}ℂ \ L̃_{q} & ⟶ & ℂ \ \overline{𝔻} \\ ν & ⟼ & (ψ_{L_{q}}(ν))^{q+1} \end{matrix}

is a (q + 1)-covering map with the asymptotic behavior as ν → ∞ given by ν ↦ k̃^{q+1}ν^{q+1}. The proposition follows.

q.e.d.

Remark 3.6 From the proof we also obtain k̃ = ^{\frac{q^{\frac{q}{q+1}}}{q+1}} . Note that the radius of capacity of L̃_{q} is \frac{1}{_{k}} so it depends on q. •

Instead of proving prop. 3.4 directly, we prove the analogous proposition in the closed wedge \overline{Σ_{q}} ⊂ Λ̃_{q}.

Proposition 3.7 1. The external rays R_{L}(±\frac{1}{2(q+1)}) land at ν = 1, the root point of the 0-limb L̃_{q,0}.

2. A parameter ν ∈ \overline{Σ_{q}} belongs to the 0-wake W L̃_{q,0} if and only if z = 0 is a repelling fixed point for P̃_{q,ν} and the landing point of R_{ν}(0).

Proof : Note that if the ray R_{ν}(0) does not branch then it must land at a fixed point. The situation is stable in ν if the fixed point is repelling. It is unstable either if the fixed point is indifferent (in which case it is parabolic of multiplier 1, due to the Snail lemma, see [Mi]) or if the ray R_{ν}(0) branches at the free critical point ω̃_{ν}. The stable set is open in Λ̃_{q} and the unstable set is closed in the same space.

The ray R_{ν}(0) passes through the free critical value ṽ_{ν} if and only if

Arg (ψ̃_{ν}^{−1}(ṽ_{ν})) = 0

From eq. (2) it follows that this is equivalent to

ν ∈ R_{L}(\frac{2n − 1}{2(q + 1)}), n ∈ ℤ

i.e., restricted to the wedge \overline{Σ_{q}} we have ν ∈ R_{L}(±\frac{1}{2(q+1)}).

For ν = \frac{q+1}{q} the polynomial P̃_{ν} has a superattracting fixed point and ℝ_{+} ⊂ ℂ \ K̃_{ν}. By symmetry, R_{ν}(0) = ℝ_{+} and R_{ν}(0) lands at z = 0.

There exists ρ > 1 so that the polynomials P̃_{ν} for ν = ±ρ exp(πi/q) ∈ ∂Σ_{q} have a superattracting cycle of period two with ω̃_{ν} < 0 < ṽ_{ν}. By symmetry, R_{ν}(0) ⊂ ℝ_{+} but R_{ν}(0) does not land at z = 0.

For ν in the wedge the situation can change only if ν = 1 for which 0 is a parabolic fixed point of multiplier 1, or if ν ∈ R_{L}(±\frac{1}{2(q+1)}) for which the ray R_{ν}(0) branches at ω̃_{ν}.

q.e.d.

Remark 3.8 For completeness we add that the connectedness locus L̃_{q} can be thought of as the connectedness locus of a different family of polynomials of degree (q + 1), namely

˜P̃_{ν}(z) = z(ν + z^{q}).

Note that ˜ P̃_{ν} and P̃_{ν} are semiconjugate by the ramified covering map Π_{q}(z) = z^{q}, i.e. Π_{q} ∘ ˜P̃_{ν} = P̃_{ν} ∘Π_{q}. A polynomial ˜ P̃_{ν} is characterized by having a fixed point at 0 and q ordinary critical points arranged symmetrically around 0. Symmetrical points have symmetrical orbits, that is ˜ P̃_{ν}(e^{\frac{2π}{q} i}z) = e^{\frac{2π}{q} i} ˜P̃_{ν}(z). The symmetry is collapsed by Π_{q} when passing to P̃_{ν}. Note that a hyperbolic component Ω of L̃_{q} deserves the name hyperbolic when viewed in this other family, i.e. if ν ∈ Ω then all critical points of ˜ P̃_{ν} are attracted to attracting cycles. All our results could be obtained from the family ˜ P̃_{ν} with a different kind of surgery. •

3.2 External Rays in the 0-wake

To end this section and for later use, we are going to rename the external rays in the 0-wake so that their arguments may be read directly from the dynamical plane as it is the case for the quadratic family (compare with eq. 1 on page 13).

Observe that eq. (3) gives the following result:

If ν ∈ W L̃_{q,0} \ L̃_{q,0} then

Arg_{K_{ν}}(ṽ_{ν}) = [\frac{1}{2} + (q + 1) Arg_{L_{q}}(ν)] (mod 1)  (4)

where Arg_{K_{ν}} (resp. Arg_{L_{q}}) means the external argument with respect to K̃_{ν} (resp. L̃_{q}). Inspired by eq. (4) we define the affine map

\begin{aligned}A_{q} : & (0, 1) & ⟶ & (−\frac{1}{2(q+1)}, \frac{1}{2(q+1)}) \\ & θ & ⟼ & \frac{1}{2} + (q + 1)θ. \end{aligned}

Recall that Π_{q} : Λ̃_{q} ⟶ Λ_{q} denotes the covering map ν ↦ λ = ν^{q}. External rays of L̃_{q} are mapped by Π_{q} onto external rays of L_{q}. For our purpose, the important external rays of L_{q} are these in the 0-wake. We parametrize them by arguments in (0, 1) as follows:

R_{L_{q,0}}(θ) = Π_{q}(R_{L_{q}}(A_{q}(θ))).

We say that θ is the external argument of the ray relative to the 0-limb.

The following proposition explains why we choose this parametrization:

Proposition 3.9 Let λ ∈ L_{q,0} be a Misiurewicz point. Then, R_{λ}(θ) lands at the critical value v_{λ} = λ(−1)^{q}(\frac{q}{q+1})^{q+1} if and only if R_{L_{q,0}}(θ) lands at λ.

The proof can be copied from [DH2] Part I, p. 74-75.

4 Dynamical Characterizations

In this section, we describe the common dynamical behavior of all polynomials Q_{c} in a rational wake of M as well as that of all polynomials P_{q,λ} in the 0-wake of L_{q}. We will use the Julia sets to construct the maps in Theorem F.

4.1 Q_{c} with c in a rational wake

Let K_{c} = K_{Q_{c}}, ψ_{c} = ψ_{Q_{c}}, etc. Since Q_{c}(−z) = Q_{c}(z), the filled Julia set K_{c} is always symmetric with respect to ω = 0, the critical point. Let us denote by β_{c} the repelling fixed point where the ray R_{c}(0) lands. The other preimage of β_{c} under Q_{c}, −β_{c}, is the landing point for R_{c}(1/2). For c in Ω_{0}, the other fixed point, α_{c}, is attracting. When c leaves the main cardioid to enter a p/q-limb, α_{c} becomes repelling and a “pinching” point in the Julia set.

Fix p/q. For each c ∈ M_{p/q}, in fact for all c in the wake W M_{p/q}, there are q rays landing at α_{c}. Their arguments are fixed throughout the wake and form a period q cycle under doubling, with combinatorial rotation number p/q. As a consequence they are rational numbers which can be written with denominators 2^{q} − 1 and numerators depending on p (see [GM]).

The other preimage of α_{c} under Q_{c} is the point α′_{c} = −α_{c}. There are q additional rays landing at α′_{c}, and their arguments are preimages under doubling of the arguments of the rays landing at α_{c}. Figs. 17 and 18 show examples of Julia sets in the 1/3 and 2/5 limbs respectively, together with the rays described above.

Figure 17: The filled Julia set for the center of the main hyperbolic component in M_{1/3} and the five subsets of the plane. The dots correspond to the period three orbit of the critical point ω = 0.

The rays landing at α_{c} and α′_{c} partition the dynamical plane into 2q − 1 closed subsets. We denote the subset containing the critical point by V_{c}^{0}, and the others by V_{c}^{i} or Ṽ_{c}^{i} = −V_{c}^{i} for i = 1, 2, … q − 1 as shown in Figs. 17 and 18. Note that these subsets have

Figure 18: The filled Julia set for the center of the main hyperbolic component in M_{2/5} and the nine subsets of the plane. The dots correspond to the period five orbit of the critical point ω = 0.

their counterparts in the right half plane for which we will not use the subscript c. For 0 ≤ i ≤ q−1 we let θ^{i} ∈ (0, 1) be the argument of the ray on the common boundary of V_{c}^{i} and V_{c}^{[i+1 (mod q)]}. In the same fashion, θ̃^{i} denotes the argument of the ray R_{c}(θ̃^{i}) = −R_{c}(θ^{i}). Note that R_{c}(θ^{i}) = R_{c}(θ̃^{i} + 1/2).

Let V_{c}^{0...q−1} = ⋃_{i=0}^{q−1} V_{c}^{i}. Then, Q_{c} acts on these sets as follows:

\begin{matrix}V_{c}^{0} & ⟶^{2−1} & Ṽ_{c}^{p} \\ V_{c}^{i}, Ṽ_{c}^{i} & ⟶^{1−1} & Ṽ_{c}^{[i+p (mod q)]} & for 0 < i < q − 1, i ≠ q − p \\ V_{c}^{q−p}, Ṽ_{c}^{q−p} & ⟶^{1−1} & V_{c}^{0...q−1} \end{matrix}  (5)

We establish the following conventions: in the dynamical plane and in expressions with integer indices like [i + p(mod q)] we will omit (mod q), while in expressions with arguments, we will omit (mod 1). In both cases, it should be understood that expressions should be taken (mod q) and (mod 1) respectively.

4.1.1 Sectors

In this part assume c ∈ M_{p/q}. For later purposes, we need to define some subsets which we call sectors. They should be viewed as neighborhoods of rays R_{c}(θ) that land.

Instead of viewing the sectors in the dynamical plane, it is better to think about them in the exterior of the unit disk or, even better, in the right half plane (see Fig. 19).

Definition For a fixed s > 0, we define the sector centered at R(θ) with slope s as :

S(θ) = S^{s}(θ) = \{ρ + 2πit ∈ \overline{ℍ} | |t − θ| ≤ sρ\}.

The boundaries of the sector S(θ) are straight lines of slope ±2πs which cross exactly at 2πiθ (see Figure 19). The doubling map M_{2} maps each sector S(θ) homeomorphically onto S(2θ).

On the dynamical plane of Q_{c} we define

S_{c}(θ) = S_{c}^{s}(θ) = ψ_{c}(exp(S(θ)).

Figure 19: Sketch of a sector in the right half plane and its correspondent in the dynamical plane.

We will only need to consider the sectors S(θ̃^{i}) and their preimages under M_{2} which, in the dynamical plane, correspond to sectors around the rays landing at α_{c} and their preimages. Since these rays land, S_{c}(θ) includes the landing point R_{c}^{∗}(θ).

Note that, as they are defined, any two sectors in ℍ overlap and hence they also overlap in ℂ \ K_{c}. To avoid this we restrict ourselves to a vertical strip as follows.

Choose η > 0 and let W_{n} denote the vertical strip in ℍ with real part less than or equal to \frac{η}{2^{n}}. That is

W_{n} = \{ρ + 2πit ∈ ℍ | ρ ≤ \frac{η}{2^{n}} \},

where n ∈ ℕ ∪ {0}. In dynamical plane, define the filled level set for the potential function as

W_{c,n} = ψ_{c}(exp(W_{n})) ∪ K_{c}.

Note that M_{2} maps W_{n+1} onto W_{n} homeomorphically while, in the dynamical plane, the polynomial Q_{c} maps every W_{c,n+1} onto W_{c,n} with degree 2.

We remark that this construction depends on the parameter c only through the function ψ_{c}. Hence, sectors are defined in the same way for all parameter values.

Figure 20: Examples of relevant sectors in the right half plane for the 1/3-limb and their correspondents in the dynamical plane.

Set S_{n}^{s}(θ) = S^{s}(θ) ∩ \overline{W}_{n}. The following proposition assures that the sectors landing at α_{c} and their preimages do not overlap, if the slope s is chosen sufficiently small (see Fig. 21).

Proposition 4.1 Fix q ∈ ℕ, η > 0 and 0 < s < \frac{1}{2η(2^{q}−1)}.

1. For each n ∈ ℕ ∪ {0} the n-sectors

S_{n}^{s}(\frac{m}{2^{n}(2^{q} − 1)}), m ∈ ℤ

are disjoint.

2. The sectors

S_{n}^{s}(\frac{m}{2^{n}(2^{q} − 1)}) m ∈ ℤ, m odd, n ∈ ℕ ∩ \{0\}

are disjoint.

Proof : The second statement follows from the first. The first statement is proved by induction on n. It is true for n = 0 since the lines

t = sρ + \frac{l}{2^{n}(2^{q} − 1)} and t = −sρ + \frac{l + 1}{2^{n}(2^{q} − 1)}

do not intersect in W_{0}.

Suppose it is true for n ≥ 0. Since

M_{2} [⋃_{m} S_{n+1}^{s}(\frac{m}{2^{n+1}(2^{q} − 1)})] = ⋃_{m} S_{n}^{s}(\frac{m}{2^{n}(2^{q} − 1)})

it is true for n + 1.

q.e.d.

In figure 21 the n-sectors and the n + 1 sectors are drawn with slope s = \frac{1}{2η(2^{q}−1)}. Two adjacent n-sectors intersect at the vertical line ρ = \frac{η}{2^{n}}.

Figure 21: n-sectors and (n + 1) sectors drawn with slope s = \frac{1}{2η(2^{q}−1)}.

4.2 P_{q,λ} with λ in the 0-wake

In this section we characterize the dynamical behavior of polynomials P_{q,λ} with λ in the 0-limb L_{q,0} or, more generally, polynomials in the 0-wake W L_{q,0}.

Recall from sect. 3.1 that polynomials P_{q,λ} with λ in W L_{q,0} are characterized by the fixed ray R_{λ}(0) landing at z = 0. Since −q is the only preimage of 0 other than itself, the preimages of R_{λ}(0)

R_{λ}(\frac{1}{q + 1}), R_{λ}(\frac{2}{q + 1}), …, R_{λ}(\frac{q}{q + 1})

land at the critical point −q, independently of λ.

These rays divide ℂ into q closed subsets. Let us denote these sets by V_{λ}^{0}, V_{λ}^{1}, …, V_{λ}^{q−1} as shown in Figure 22. We will always draw the dynamical plane for P_{q,λ} rotated by 180^{∘}. The reason will become clear in sect. 5.2. The polynomial P_{q,λ} maps each of these subsets as follows:

\begin{matrix}V^{∘} _{λ}^{0} & ⟶^{2−1} & ℂ \ (R_{λ}(0) ∪ \{0\}) \\ V^{∘} _{λ}^{i} & ⟶^{1−1} & ℂ \ (R_{λ}(0) ∪ \{0\}) for 1 ≤ i ≤ q − 1 \end{matrix}  (6)

As in section 4.1.1, for θ ∈ ℝ let S(θ) = S^{s}(θ) denote a sector in \overline{ℍ} with slope s. The map M_{q+1} (multiplication by (q+1)) maps S(θ) homeomorphically onto S((q+1)θ). When

Figure 22: The filled Julia set for the center of the main hyperbolic component in L_{3,0} (left) and L_{5,0} (right) and the three (resp. five) subsets of the plane. In both cases ω is fixed. The planes have been rotated 180 degrees.

λ ∈ L_{q,0} we may transport these sectors to the dynamical plane by defining

S_{λ}(θ) = ψ_{λ}(exp(S(θ))).

To avoid the overlapping, fix η > 0 and let W_{n} be as above, substituting \frac{η}{2^{n}} by \frac{η}{(q+1)^{n}}. Let also W_{q,λ,n} be the filled level set for the potential function, i.e.

W_{q,λ,n} = ψ_{λ}(exp(W_{n})) ∪ K_{λ}

We will consider only the sector centered at the 0-ray and its preimages (see Fig. 23). As in section 4.1.1, one can show that if 0 < s < \frac{1}{2η(q+1)}, these sectors do not overlap for ρ < η.

5 Surgery: Proof of Theorem F

5.1 Idea of the Proof

We prove Theorem F using surgery following the methods in [BD]. In this section we sketch the steps of the proof .

Let p and q be positive numbers, p < q, q ≥ 3 and gcd(p, q) = 1. Then, for any value c ∈ M_{p/q} we will assign a value λ ∈ L_{q,0}.

Figure 23: Sketch of relevant sectors in the right half plane for any λ ∈ L_{3,0} and their correspondents in dynamical plane.

We start in the dynamical plane of Q_{c}. Through cutting and sewing, we construct a new map f_{c}^{(1)}, the first return map. This map has several lines of discontinuity.

Next we smooth f_{c}^{(1)} on sectors and obtain a new map f_{c}^{(2)} which is quasi-regular and has two critical points: one with multiplicity q − 1 and the other with multiplicity one.

To obtain an analytic map, we define an f_{c}^{(2)}-invariant almost complex structure, σ. By the Measurable Riemann Mapping Theorem, this structure can be integrated by a quasi-conformal homeomorphism φ_{c}, inducing the analytic map f_{c}^{(3)} = φ_{c} ∘ f_{c}^{(2)} ∘ φ_{c}^{−1}.

We then show that f_{c}^{(3)} is a polynomial-like map (see sect. 2.2 or [DH3]) of degree q + 1. By the Straightening Theorem, it is hybrid equivalent to a polynomial P of degree q + 1. Finally, we show that P may be chosen to belong to the family L_{q,0}, and that it is independent of the choices made in the above steps.

This process defines the map ϕ_{p/q} : M_{p/q} ⟶ L_{q,0}. In Sec. 5.2, we explain the construction in the dynamical plane. In Sect. 5.3, 5.4 and 5.5 we prove that ϕ_{p/q} is bijective, analytic in the interior of M_{p/q} and continuous at points in the boundary of M_{p/q}, respectively.

5.2 Construction of a polynomial of degree q + 1. Definition of ϕ_{p/q}.

We start in the dynamical plane of a polynomial Q_{c} with c ∈ M_{p/q}, for a fixed p/q, and with W_{c}, V_{c}^{i}, S_{c}(θ), etc. as in section 4.1. Let V_{c}^{0...q−1} = ⋃_{i=0}^{q−1} V_{c}^{i}. Our goal is to obtain a value λ = λ(c) ∈ L_{q,0}.

We construct a truncated space ℂ_{c}^{T} obtained from V_{c}^{0...q−1} by identifying

R_{c}(θ̃^{0}) with R_{c}(θ̃^{q−1})

equipotentially (see Figs. 24 and 25), that is, we identify a point z ∈ R_{c}(θ̃^{0}) with the unique point w ∈ R_{c}(θ̃^{q−1}) such that G_{c}(z) = G_{c}(w). Throughout the rest of the section we let θ̃ denote this identified argument.

Figure 24: Left:The space ℂ_{c}^{T} where a random equipotential has been drawn and its preimage under f_{c}^{(1)}, for c ∈ M_{1/3}. Right: The space ℍ^{T} with a vertical line and its preimage under f^{(1)}, when working in the 1/3-limb.

Figure 25: Left:The space ℂ_{c}^{T} where a random equipotential has been drawn and its preimage under f_{c}^{(1)}, for c ∈ M_{2/5}. Right: The space ℍ^{T} with a vertical line and its preimage under f^{(1)}, when working in the 2/5-limb.

Note that the space ℂ_{c}^{T} can be viewed as the quotient of V_{c}^{0...q−1} union some neighborhoods of the rays R_{c}(θ̃^{q−1}) and R_{c}(θ̃^{0}) by the equivalence relation identifying the points in the two neighborhoods

ψ_{c}(exp(ρ + 2πi(θ̃^{0} + t))) ∼ ψ_{c}(exp(ρ + 2πi(θ̃^{q−1} + t))).

Thus, ℂ_{c}^{T} is a Riemann surface isomorphic to ℂ.

Define the truncated filled Julia set as

K_{c}^{T} = K_{c} ∩ ℂ_{c}^{T} = K_{c} ∩ V_{c}^{0...q−1}.

Note that no identification takes place in K_{c}, so K_{c}^{T} can be viewed as a subset of K_{c}.

We define now a map f_{c}^{(1)} : ℂ_{c}^{T} ⟶ ℂ_{c}^{T}, the first return map, so that f_{c}^{(1)} acts on the sets V^{∘} _{c}^{i} similarly to the polynomial P_{q,λ} with λ ∈ L_{q,0} (compare with (6) on page 30). That is we define

f_{c}^{(1)} = \begin{cases}Q_{c}^{q} & if z ∈ V^{∘} _{c}^{0}, \\ Q_{c}^{q−1} & if z ∈ V^{∘} _{c}^{p} \\ Q_{c}^{q−2} & if z ∈ V^{∘} _{c}^{2p} \\ … \\ Q_{c} & if z ∈ V^{∘} _{c}^{(q−1)p} = V^{∘} _{c}^{q−p} \\ α_{c} & if z = α′_{c} \end{cases}

Figure 26: Left:The space ℂ_{c}^{T} and the domain and the image of f_{c}^{(2)}, for c ∈ M_{1/3}. Right: The space ℍ^{T} and the domain and the image of f^{(2)} when working in the 1/3-limb.

Note that f_{c}^{(1)} is well defined on the identified ray R_{c}(θ̃) and holomorphic everywhere except at α′_{c} and at

R′_{c} = R_{c}(θ^{0}) ∪ R_{c}(θ^{1}) ∪ ⋯ ∪ R_{c}(θ^{q−1})

where it is discontinuous.

Informally, just to get some intuition, note as well that the first return map already has some of the characteristics of a map of the family P_{q,λ}. For example, each of the subsets V^{∘} _{c}^{i} is mapped under f_{c}^{(1)} to the space ℂ_{c}^{T} \ (R_{c}(θ̃) ∪ {α_{c}}). The map has degree two in V^{∘} _{c}^{0}, and degree one on the other subsets V^{∘} _{c}^{i}, which justifies the hope that some variation of the first return map will be a map of degree q + 1. Also, we can observe the creation of a new critical point, at α′_{c}. This is still a prefixed point, but a neighborhood around it now wraps q times around α_{c} under f_{c}^{(1)} , making α′_{c} a critical point of multiplicity q − 1.

This ends the first step of the construction. We will now modify the map f_{c}^{(1)} to construct a new map f_{c}^{(2)} which will be quasi-regular. The modification is done only on neighborhoods of the rays (the sectors defined in section 4.1.1) for which f_{c}^{(1)} is discontinuous. We denote these sectors by S′_{c}, i.e.,

S′_{c} = S_{c}(θ^{0}) ∪ S_{c}(θ^{1}) ∪ ⋯ ∪ S_{c}(θ^{q−1}).

We will have to restrict the space so that these sectors do not overlap. We will do this construction once and for all in the right half plane.

We define ℍ^{T} from ℍ by cutting along R(n+θ̃^{0}) and R(n+θ̃^{q−1}), n ∈ ℤ and identifying the two rays as we did for ℂ_{c}^{T}. Then, the universal covering map ψ_{c} ∘ exp : ℍ ⟶ ℂ \ K_{c} restricts to a universal covering map

(ψ_{c} ∘ exp)^{T} : ℍ^{T} ⟶ ℂ_{c}^{T} \ K_{c}^{T}

which is holomorphic in both variables.

Let R′, S′, S and V^{i} denote the sets in ℍ^{T} that project to R′_{c}, S′_{c}, S_{c}(θ̃) and V_{c}^{i} respectively.

Choose a lift f^{(1)} of f_{c}^{(1)}. Then, the following diagram commutes

ℍ^{T}  ⟶^{f^{(1)}} ℍ^{T}
(ψ_{c}∘exp)^{T} ↓  ↓^{(ψ_{c}∘exp)^{T}}
ℂ_{c}^{T} \ K_{c}^{T} ⟶^{f_{c}^{(1)}} ℂ_{c}^{T} \ K_{c}^{T}

Note that f^{(1)} can be chosen independently of c and that one can actually write down such a map explicitly. The map f^{(1)} is holomorphic everywhere except at R(θ^{i}+n), 0 ≤ i ≤ q−1, n ∈ ℤ where it is discontinuous.

We proceed now to restrict the domain. We will define two sets X′ and X which will be the domain, respectively the image, of the new map f^{(2)}. Set X = W ∩ ℍ^{T} and let γ denote the curve in ℍ^{T} which bounds X. Let γ′ denote a C^{∞} curve which projects to a Jordan curve γ′_{c} = (ψ_{c} ∘ exp)^{T} (γ′) in ℂ_{c}^{T} and which equals

Re(γ′) = \begin{cases}\frac{η}{2^{q}} & if z ∈ V^{0} \ S′ \\ \frac{η}{2^{q−1}} & if z ∈ V^{p} \ S′ \\ … \\ \frac{η}{2} & if z ∈ V^{(q−1)p} \ S′ = V^{q−p} \ S′ \end{cases}

Let X′ be the vertical strip bounded by γ′ (see Fig. 26). In dynamical plane, set X_{c} = W_{c} ∩ ℂ_{c}^{T} and let X′_{c} denote the filled set in ℂ_{c}^{T} bounded by γ′_{c}.

We shall modify the map f^{(1)} on the sectors around the rays of discontinuity as shown in Fig. 26 so that it induces a quasi-regular map

f_{c}^{(2)} : X′_{c} ⟶ X_{c}

First we divide the sectors in S′ into quadrilaterals as described below.

For S(θ) ⊂ (S′ ∪ S) we have a homeomorphism

Π_{θ} : S(θ) ⟶ S(θ)

which is multiplication by 2^{q} followed by a vertical translation that depends on θ. For S(θ) ⊂ S we define

T_{0}(θ) = \overline{(S(θ) ∩ X) \ Π_{θ}^{−1}(S(θ) ∩ X)}  (7)

and for k > 0

T_{k}(θ) = Π_{θ}^{−1}(T_{k−1}(θ)).  (8)

For S(θ) ⊂ S′, define

T_{0}(θ) = \overline{(S(θ) ∩ X′) \ Π_{θ}^{−1}((S(θ) ∩ X′))}

and T_{k}(θ) as above.

Note that the map Π_{θ} restricted to T_{k}(θ) is a conformal isomorphism from T_{k}(θ) to T_{k−1}(θ).

We define f^{(2)} : X′ ⟶ X to equal f^{(1)} outside of S′. On these sectors, f^{(2)} is defined by induction as follows. Let n_{i} denote the integer determined by

f^{(1)} : ∂(S(θ^{i})) ⟶ ∂(S(n_{i} + θ̃)).

Then choose for 0 ≤ i ≤ q − 1 a diffeomorphism

f_{0,i}^{(2)} : T_{0}(θ^{i}) ⟶ T_{0}(n_{i} + θ̃)

so that f^{(1)} and f_{0,i}^{(2)} determine the same tangent map on the boundary of the sectors. For k > 0 define

f_{k,i}^{(2)} : T_{k}(θ^{i}) ⟶ T_{k}(n_{i} + θ̃)

by induction as the map satisfying

Π_{n_{i}+_{θ}} ∘ f_{k,i}^{(2)} = f_{k−1,i}^{(2)} ∘ Π_{θ^{i}}.

Finally extend f^{(2)} : X′ ⟶ X as a covering transformation so it is compatible with the projection. The map f_{c}^{(2)} : X′_{c} ⟶ X_{c} is defined on ℂ_{c}^{T} \ K_{c}^{T} by

X′  ⟶^{f^{(2)}} X
(ψ_{c}∘exp)^{T} ↓  ↓^{(ψ_{c}∘exp)^{T}}
X′_{c} \ K_{c}^{T} ⟶^{f_{c}^{(2)}} X_{c} \ K_{c}^{T}

and f_{c}^{(2)}|_{K_{c}^{T}} = f_{c}^{(1)}|_{K_{c}^{T}}.

The map f^{(2)} : X′ ⟶ X is a diffeomorphism. The map f_{c}^{(2)} : X′_{c} ⟶ X_{c} is a degree (q + 1)-ramified covering.

Lemma 5.1 The map f^{(2)} is quasi-conformal. Hence f_{c}^{(2)} is quasi-regular.

Proof : We must show that the field of ellipses E_{x} = (T_{x}f^{(2)})^{−1}(S^{1}) for x ∈ X′, has bounded dilatation ratio.

If x ∉ S′, then E_{x} is a circle, since the map f^{(2)} outside S′ equals f^{(1)} which is analytic. If x ∈ T_{0}(θ^{i}) for some i, then the ratio of the axes of E_{x} is bounded by some constant

K, since f_{0,i}^{(2)} is a diffeomorphism and T_{0}(θ^{i}) is compact.

If x ∈ T_{k}(θ^{i}) for some k > 0 then the ratio of the axes of E_{x} is bounded by the same constant K, since f_{k,i}^{(2)} is f_{0,i}^{(2)} composed with a finite number of analytic maps.

The same can be said for points that belong to the integer translations of this sectors since f^{(2)} is a covering transformation.

Hence, the dilatation ratio of the field of ellipses (E_{x})_{x∈X′} is bounded by the same constant for all x ∈ X′, and f^{(2)} is quasi-conformal. Since (ψ_{c} ∘ exp)^{T} is conformal, f_{c}^{(2)} is quasi-regular.

q.e.d.

{Remarks 5.2 \atop 1.} It is important to note that orbits enter S′ at most once. This allows us to bound the dilatation ratio by the same constant everywhere, since the diffeomorphism f_{0,i}^{(2)} has to be applied only once. Moreover it is essential for the next step in the construction when we change the complex structure. This is the Shishikura principle for surgery of this type.

2. The map f_{c}^{(2)} is a degree (q + 1) ramified covering. It has two critical points: one at ω = 0 with multiplicity one and another one at α′_{c} with multiplicity q − 1, mapped to the fixed point α_{c}. These are the topological characteristics of maps in the family P_{q,λ}, but the map f_{c}^{(2)} is not holomorphic.

The next step (holomorphic smoothing) is to construct a map f_{c}^{(3)} which has the same properties as f_{c}^{(2)} but so that it also is a polynomial-like map of degree q + 1. In order to do so, we change the complex structure into a new almost complex structure with the property that it is preserved under the map f_{c}^{(2)} .

Lemma 5.3 There exists an almost complex structure σ_{c} on X_{c} quasi-conformally equivalent to the standard complex structure σ_{0}, such that (f_{c}^{(2)})^{∗}σ_{c} = σ_{c} and σ_{c} = σ_{0} on the set K_{c}^{T}.

This lemma could also be phrased as follows: There exists a measurable field of ellipses in X_{c} with bounded ratio of the axes. These ellipses are invariant under f_{c}^{(2)} and the ellipses are circles on K_{c}^{T}. As before, we will construct this almost complex structure in the right half plane. Proof : For x ∈ S′, we define the complex structure σ to be the same as in lemma 5.1, i.e.

E_{x} = (T_{x}f^{(2)})^{−1}(S^{1}),

and then, the ratio of the axes is bounded by some constant K.

If x ∉ S′, then either there exists n > 0 such that (f^{(2)})^{n}(x) ∈ S′ or the orbit of x never enters S′.

If there exists such n, then it is unique since a point in S′ is mapped to a point in S and does not leave S as long as the map is defined. In this case define

E_{x} = (T_{x}(f^{(2)})^{n})^{−1}(E_{(f^{(2)})^{n}(x)}).

Since f^{(2)} is holomorphic outside of the sectors in S′, we have that (f^{(2)})^{n} is holomorphic in a neighborhood of x. Hence, the dilatation ratio of E_{x} is the same as the dilatation ratio of E_{(f^{(2)})^{n}(x)}, and therefore bounded by K.

Finally, if the orbit of x never enters S′, define E_{x} = S^{1}.

By construction, σ is invariant under f^{(2)}. Define σ_{c} on X_{c} \ K_{c}^{T} as the pull-back of σ by the map ((ψ_{c} ∘ exp)^{T})^{−1} on a fundamental domain, and set σ_{c} = σ_{0} on K_{c}^{T}. By construction, σ_{c} is the required almost complex structure.

q.e.d.

By doing this construction once and for all in the right half plane and pulling back by (ψ_{c} ∘ exp)^{T} we have that σ_{c} and f_{c}^{(2)} vary holomorphically with respect to c.

Let σ_{c} be the almost complex structure given by Lemma 5.3. We now apply the Measurable Riemann Mapping Theorem (see Sect. 2.2) on X^{∘}_{c}, a Riemann surface isomorphic to 𝔻, to obtain a unique quasi-conformal homeomorphism φ_{c} :X^{∘}_{c}⟶ 𝔻 integrating σ_{c} and satysfying φ_{c}(ω_{c}) = 0 and φ_{c}(α_{c}) ∈ ℝ_{+}. We have

(T_{x}φ_{c})^{−1}(S^{1}) = ρ_{c}(x)E_{x}

where ρ_{c}(x) ∈ ℝ_{+}.

Now set D′_{c} = φ_{c}(X^{∘}′_{c}) ⊂ 𝔻 and define f_{c}^{(3)} by

X^{∘}′_{c} ⟶^{f_{c}^{(2)}} X^{∘}_{c}
φ_{c}↓  ↓^{φ_{c}}
D′_{c} ⟶^{f_{c}^{(3)}} 𝔻

Remark 5.4 In this particular case, the quasi-conformal homeomorphism φ_{c} is actually a hybrid equivalence, since σ_{c} = σ_{0} on K_{c}^{T}. •

We show that f_{c}^{(3)} is holomorphic as follows: φ_{c}^{−1} takes the standard structure σ_{0} of U′_{c} to the almost complex structure σ_{c}. Then, the map f_{c}^{(2)} preserves σ_{c} which is taken back to σ_{0} by φ_{c}. We conclude that f_{c}^{(3)} takes σ_{0} to σ_{0} and therefore is holomorphic.

Since φ_{c} is a homeomorphism, f_{c}^{(3)} is still a ramified covering of degree q + 1 with two critical points (see remarks 5.2). Since D′_{c} is relatively compact in 𝔻, we can conclude that f_{c}^{(3)} is a polynomial-like map of degree q + 1.

We now apply the Straightening Theorem to obtain a polynomial P_{q,λ(c)} of degree q + 1 in L_{q,0}, and a hybrid equivalence χ_{c} that conjugates f_{c}^{(3)} to P_{q,λ(c)} on neighborhoods of K_{c}^{T} and K_{λ(c)}.

To justify this claim notice that the polynomial χ_{c} ∘ f_{c}^{(3)} ∘ χ_{c}^{−1} satisfies the following properties:

• the point z_{1} = χ_{c}(φ_{c}(α_{c})) is a repelling fixed point,

• the point z_{2} = χ_{c}(φ_{c}(0)) is a critical point of multiplicity one,

• the point z_{3} = χ_{c}(φ_{c}(α′_{c})) is a critical point of multiplicity q − 1, • z_{2} is mapped to z_{1}, and

• z_{1} is the landing point of one fixed ray.

Since we may conjugate by an affine transformation sending z_{1} to 0 and z_{3} to −q, we may assume that the polynomial is of the form P_{q,λ(c)} .

This concludes the construction on the dynamical plane. By this construction, given a parameter value c ∈ M_{p/q} we have obtained a unique λ(c) ∈ L_{q,0}. We define the map

ϕ_{p/q} : M_{p/q} ⟶ L_{q,0}

by ϕ_{p/q}(c) = λ(c).

{Remarks 5.5 \atop 1.} The value of λ does not depend on the choices made throughout the construction, i.e. the choice of the boundaries of X and X′, the slope s, the diffeomorphisms f_{0,i}^{(2)} and the integrating map φ_{c}. Indeed, suppose that by other choices η̃, γ̃′, s̃, f̃_{0,i}^{(2)} and φ̃_{c} we obtain a map f̃^{(2)} and a polynomial P_{q,_{λ}}. Then, f_{c}^{(2)} and f̃_{c}^{(2)} are hybrid equivalent and also P_{q,λ} and P_{q,_{λ}} would be hybrid equivalent (because of remark 5.2). Hence by

Prop. 3.3 we have λ = λ̃.

2. Following the construction of the Straightening map in [DH3], one can check that χ_{c} can be defined in all of φ_{c}(X_{c}), since the boundary of X_{c} is smooth and φ_{c} is quasi-conformal.

The following proposition addresses the question of uniqueness of the conjugating maps.

Proposition 5.6 Suppose c ∈ M_{p/q} and λ = ϕ_{p/q}(c) ∈ L_{q,0}. A hybrid equivalence between f_{c}^{(3)} and P_{λ} is uniquely determined on φ_{c}(K_{c}^{T}), hence χ_{c} ∘ φ_{c} is uniqueley determined on K_{c}^{T}.

Proof : Suppose χ_{c} and χ′_{c} are hybrid equivalences between f_{c}^{(3)} and P_{λ}. Then φ = χ′_{c}∘χ_{c}^{−1} is a hybrid equivalence between P_{λ} and itself. By copying the proof of Prop. 6 in [DH3], p. 302 defining

Φ = \begin{cases}φ & on K_{λ} \\ Id & on ℂ \ K_{λ} \end{cases}

we obtain that Φ is holomorphic and hence the identity on all of ℂ.

q.e.d.

The following theorem is the dynamical counterpart of Theorem F.

Theorem G

Suppose c ∈ M_{p/q} and λ = ϕ_{p/q}(c) ∈ L_{q,0}. Recall that X_{c} is a neighborhood of K_{c}^{T} bounded by the truncated equipotential γ_{c} and that W_{λ} is a neighborhood of K_{λ} namely the filled level set of a chosen equipotential of G_{λ}. There exists a homeomorphism

H_{c} : X_{c} ⟶ W_{λ}

which satisfies \overline{∂}H_{c} = 0 on K_{c}^{T} and conjugates f_{c}^{(2)} to P_{λ} and such that H_{c} maps relevant sectors and rays for Q_{c} to their counterparts for P_{λ}, i.e. H_{c} maps S_{c}(θ̃), R_{c}(θ̃), and R_{c}(θ^{i}) to S_{λ}(0), R_{λ}(0), and R_{λ}(\frac{i+1}{q+1}) respectively.

Moreover, H_{c}|_{K_{c}^{T}} is uniquely determined and it conjugates the first return map f_{c}^{(1)} to P_{λ}.

Proof : The map

χ_{c} ∘ φ_{c} : X_{c} → χ_{c}(φ_{c}(X_{c}))

is a quasi-conformal homeomorphism from X_{c} to a neighborhood of K_{λ} conjugating by construction the map f_{c}^{(2)} to the polynomial P_{λ}. In particular, H_{c} conjugates the first return map f_{c}^{(1)} on K_{c}^{T} to the polynomial P_{λ} on K_{λ}. By remark 5.2, it satisfies \overline{∂}H_{c} = 0 on K_{c}^{T}.

The map χ_{c} ∘ φ_{c} sends S_{c}(θ̃), R_{c}(θ̃), R_{c}(θ^{i}) and γ_{c} to some quasi-conformal image of these objects. We claim that there exists a hybrid equivalence h_{λ} : χ_{c}(φ_{c}(X_{c})) ⟶ W_{λ} conjugating P_{λ} to itself and such that H_{c} := h_{λ} ∘ χ_{c} ∘ φ_{c} satisfies the properties in the theorem. In the remainder of this proof we give the idea of the construction of h_{λ}.

Let 𝔄 and A denote the annuli

𝔄 = \overline{χ_{c}(φ_{c}(X_{c})) \ P_{λ}^{−1}(χ_{c}(φ_{c}(X_{c})))}
A = \overline{W_{λ} \ W_{λ,1}}.

First define h_{λ} : 𝔄 ⟶ A as a diffeomorphism conjugating P_{λ} to itself on the inner boundaries and sending χ_{c}(φ_{c}(S_{c}(θ̃))) ∩ 𝔄, χ_{c}(φ_{c}(R_{c}(θ̃))) ∩ 𝔄 and χ_{c}(φ_{c}(R_{c}(θ^{i}))) ∩ 𝔄 to S_{λ}(0) ∩ A, R_{λ}(0) ∩ A and R_{λ}(\frac{i+1}{q+1}) ∩ A respectively (see Fig. 27). This can easily be done constructing first such a map in the right half plane.

Once h_{λ} is correctly defined on the annulus we use a pull-back argument to define it everywhere else outside K_{λ}, that is if we denote 𝔄^{n} = P_{λ}^{−n}(𝔄) and A^{n} = P_{λ}^{−n}(A), we define h_{λ} = P_{λ}^{−n} ∘ h_{λ} ∘ P_{λ}^{n} on 𝔄^{n}. Finally, define h_{λ} to be the identity on K_{λ}. By an argument analogous to Lemma 1 in page 302 of [DH3], followed by Rickman’s lemma in page 303, one can show that h_{λ} is a hybrid equivalence between P_{λ} and itself with the required properties.

That H_{c}|_{K_{c}^{T}} is uniquely defined follows from the proposition above.

q.e.d.

Figure 27: The annuli 𝔄 and A and the function h_{λ}.

{Remarks 5.7 \atop 1.} If c is Misiurewicz, i.e. if ω_{c} = 0 is strictly preperiodic for Q_{c}, then ω_{λ} = \frac{−q}{q+1} is strictly preperiodic for P_{λ}. In particular if ω_{c} is eventually mapped to α_{c} then ω_{λ} is eventually mapped to the fixed point 0.

2. If c is hyperbolic, i.e. if ω_{c} is attracted to an attracting cycle by Q_{c}, then ω_{λ} is attracted to an attracting cycle by P_{λ}. Moreover, the multiplier of the cycle by Q_{c} equals the multiplier of the cycle by P_{λ}, since H_{c} is holomorphic in the interior of K_{c}^{T}.

3. The main branch point (Misiurewicz) in the p/q-limb of M corresponds to Q_{c}^{q}(ω_{c}) = α′_{c}. It follows that the main branch point in L_{q,0} corresponds to the polynomial P_{q,λ}(ω_{λ}) = −q or λ = (q + 1)(1 + \frac{1}{q})^{q}. Hence this branch point tends to ∞ as q tends to ∞.

5.3 Bijectivity of ϕ_{p/q}

In this section, we construct a map

ξ_{p/q} : L_{q,0} ⟶ M_{p/q}.

and show that it is the inverse of ϕ_{p/q}.

We start with a fixed λ ∈ L_{q,0} and our goal is to obtain a unique quadratic polynomial Q_{c} such that c ∈ M_{p/q}. Note that this means that, from each L_{q,0}, we have to construct several maps, one for each p.

Let V_{λ}^{i}, etc. be defined as in Sect. 4.2. Since this construction is supposed to be the opposite of the construction of Sect. 5.2, we first want to “add” the piece of dynamical plane that we removed before.

Figure 28: Left: The filled Julia set for P_{5,λ} where λ is the center of a hyperbolic component of period two in L_{5,0}; λ is the image of both c-values in Fig. 3 under ϕ_{1/5} and ϕ_{2/5} respectively. Right: The Hubbard tree (compare with Fig. 4)

Hence, we cut V_{λ}^{0} along R_{λ}(0) in such a way that now V_{λ}^{0} contains two copies of R_{λ}(0), which we denote by R_{λ}(0) and R_{λ}(1). For 1 ≤ i ≤ q − 1, let Ṽ_{λ}^{i} be an identical copy of V_{λ}^{i} and let

τ_{λ}^{i} : V_{λ}^{i} ⟶ Ṽ_{λ}^{i}

be the identity map. The copies Ṽ_{λ}^{i} are sewn together in order to resemble Figs. 17 and 18, as shown in figures 29 or 30. We glue the boundaries (the rays) together in order to construct a Riemann surface ℂ_{λ}^{E} isomorphic to ℂ. However, we will not identify the rays equipotentially as we did before. (If we did, the map that we are going to define would not be continuous on these rays). In what follows, we describe these identifications. Figs. 29 and 30 show two examples.

Let N be the bijective function that assigns to each integer 1 ≤ i ≤ q − 1 the unique integer 1 ≤ N(i) ≤ q − 1 such that

N(i) p ≡ i (mod q).

Intuitively, if this were the dynamical plane of a quadratic polynomial Q_{c} (with c ∈ M_{p/q}), N(i) would be the number of iterates that it takes for V_{λ}^{0} to be mapped to Ṽ_{λ}^{i} for the first time.

Pick 1 < i < q − 1 and suppose N(i − 1) < N(i). Then, for each point x ∈ R_{λ}(i/(q + 1))

Figure 29: The space ℂ_{λ}^{E} and the domain of f_{λ}^{(2)} for λ ∈ L_{3,0} and p = 1.

Figure 30: The space ℂ_{λ}^{E} and the domain of f_{λ}^{(2)} for λ ∈ L_{5,0} and p = 2.

of potential ρ (i.e. G_{λ}(x) = ρ), we identify

τ_{λ}^{i−1}(x) ∼ τ_{λ}^{i}(y),

where y ∈ R_{λ}(i/(q + 1)) and has potential G_{λ}(y) = ρ / (q + 1)^{\frac{N(i)−N(i−1)}{q}}.

Otherwise, if N(i − 1) > N(i), then we identify

τ_{λ}^{i}(x) ∼ τ_{λ}^{i−1}(y),

where x, y ∈ R_{λ}(i/(q + 1)), G_{λ}(x) = ρ, and G_{λ}(y) = ρ / (q + 1)^{\frac{N(i−1)−N(i)}{q}}.

If i = 1, for each x ∈ R_{λ}(1) with potential ρ we identify

x ∼ τ_{λ}^{1}(y),

where y ∈ R_{λ}(1/(q + 1)) and has potential G_{λ}(y) = ρ / (q + 1)^{\frac{N(1)}{q}}.

If i = q − 1 then for each x ∈ R_{λ}(0), we identify

x ∼ τ_{λ}^{q−1}(y),

where y ∈ R_{λ}(q/(q + 1)) and has potential G_{λ}(y) = ρ(q + 1)^{\frac{N(q−1)}{q}}. Finally, we identify the landing points of the rays bounding the Ṽ_{λ}^{i} with 0. We shall denote these q rays by R̃_{λ}(\frac{i}{q+1}), for 1 ≤ i ≤ q.

We define

ℂ_{λ}^{E} = ⋃_{i=0}^{q−1} V_{λ}^{i} ∪ ⋃_{i=1}^{q−1} Ṽ_{λ}^{i}/ ∼

with the identifications above.

Define the extension of K_{λ} to be K_{λ}^{E} = K_{λ} ∪ ⋃_{i=1}^{q−1} τ_{λ}^{i}(K_{λ})/ ∼. Note that K_{λ} can be viewed as a subset of K_{λ}^{E}.

We define

f_{λ}^{(1)} : ℂ_{λ}^{E} ⟶ ℂ_{λ}^{E}

so that f_{λ}^{(1)} acts on the sets V_{λ}^{0}, V_{λ}^{i}, Ṽ_{λ}^{i} similarly to the quadratic polynomial Q_{c} with c ∈ M_{p/q} (compare with (5) on page 27). It should be viewed as the “inverse” of the first return map.

Recall that P_{λ} : V^{∘} _{λ}^{i} ⟶ ℂ \ (R_{λ}(0) ∪ {0}) is a homeomorphism and define then

f_{λ}^{(1)} = \begin{cases}τ_{λ}^{(i+1)p} ∘ P_{λ}|_{V_{λ}^{(i+1)p}}^{−1} ∘ P_{λ} & on V_{λ}^{ip}, for 0 ≤ i ≤ q − 2 \\ P_{λ} & on V_{λ}^{q−p} \\ f_{λ}^{(1)} ∘ (τ_{λ}^{ip})^{−1} & on Ṽ_{λ}^{ip}, for 1 ≤ i ≤ q − 1 \end{cases}

Note that f_{λ}^{(1)} is well defined on the identified rays

R̃_{λ} = R̃_{λ}(\frac{1}{q + 1}) ∪ R̃_{λ}(\frac{2}{q + 1}) ∪ ⋯ ∪ R̃_{λ}(\frac{q}{q + 1})

and holomorphic everywhere except at −q and at

R_{λ} = R_{λ}(\frac{1}{q + 1}) ∪ R_{λ}(\frac{2}{q + 1}) ∪ ⋯ ∪ R_{λ}(\frac{q}{q + 1})

where it is discontinuous. As before, we shall smoothen the map on sectors and restrict the space so that the sectors do not overlap. We will do this - once and for all - in the right half plane.

We define ℍ^{E} from ℍ by cutting along R(n), for n ∈ ℤ, and gluing in copies of V^{i} with similar identifications along the rays R̃(n + \frac{i}{q+1}), 1 ≤ i ≤ q, n ∈ ℤ, as in ℂ_{λ}^{E}. The universal covering map

ψ_{λ} ∘ exp : ℍ ⟶ ℂ \ K_{λ}

extends in an obvious way to a universal covering map

(ψ_{λ} ∘ exp)^{E} : ℍ^{E} ⟶ ℂ_{λ}^{E} \ K_{λ}^{E}.

The map (ψ_{λ} ∘ exp)^{E} is holomorphic in both variables. Let S, S̃, Ṽ^{i} denote the sets in ℍ^{E} that project to ⋃_{i}S_{λ}(\frac{i}{q+1}), ⋃_{i}S̃_{λ}(\frac{i}{q+1}) and Ṽ^{i} respectively.

Choose a lift f^{(1)} of f_{λ}^{(1)}. Then the following diagram commutes:

ℍ^{E}  ⟶^{f^{(1)}} ℍ^{E}
(ψ_{λ}∘exp)^{E} ↓  ↓^{(ψ_{λ}∘exp)^{E}}
ℂ_{λ}^{E} \ K_{λ}^{E} ⟶^{f_{λ}^{(1)}} ℂ_{λ}^{E} \ K_{λ}^{E}

Note that we can choose f^{(1)} independently of λ. The map f^{(1)} is holomorphic, except at R(n + \frac{i}{q+1}), 1 ≤ i ≤ q − 1, n ∈ ℤ where it is discontinuous.

Let X ⊂ ℍ^{E} denote the vertical strip in ℍ^{E} bounded by the curve γ which equals

Re(γ) = \begin{cases}η & in V^{0...q−1} \\ \frac{η}{(q+1)^{i/q}} & in Ṽ^{ip} \end{cases}

Due to the identification of the rays, γ is a vertical line. Set γ_{λ} = (ψ_{λ} ∘ exp)^{E}(γ). Let γ′ denote a C^{∞} curve which projects to a Jordan curve

γ′_{λ} = (ψ_{λ} ∘ exp)^{E}(γ′)

in ℂ_{λ}^{E}, and which equals

Re(γ′) = \begin{cases}\frac{η}{(q+1)^{(i+1)/q}} & in V^{ip} \ S′, 0 ≤ i ≤ q − 1 \\ \frac{η}{(q+1)^{(i+1)/q}} & in Ṽ^{ip}, 1 ≤ i ≤ q − 1 \end{cases}

Let X′ be the vertical strip in ℍ^{E} bounded by γ′ and let X_{λ}, respectively X′_{λ} denote the filled sets in ℂ_{λ}^{E} of γ_{λ}, respectively γ′_{λ}.

We shall modify the map f^{(1)} on the sectors around the rays of discontinuity so it induces a quasi-regular map

f_{λ}^{(2)} : X′_{λ} ⟶ X_{λ}.

For θ = n + \frac{i}{q+1}, n ∈ ℤ, 0 ≤ i ≤ q we have a homeomorphism

Π_{θ} : S(θ) ⟶ S(θ)

which is multiplication by q + 1 composed with a vertical translation.

For the same values of θ, the maps Π_{θ} induce homeomorphisms

Π̃_{θ} : S̃(θ) ⟶ S̃(θ).

Define

T̃_{0}(θ) = \overline{(S̃(θ) ∩ X) \ Π̃_{θ}^{−1}(S̃(θ) ∩ X)},

and for k > 0

T̃_{k}(θ) = Π̃_{θ}^{−1}(T̃_{k−1}(θ)).

For θ = n + \frac{i}{q+1}, n ∈ ℤ, 1 ≤ i ≤ q, define

T_{0}(θ) = \overline{(S(θ) ∩ X′) \ Π_{θ}^{−1}(S(θ) ∩ X′)},

and T_{k}(θ) as above.

We define f^{(2)} : X′ ⟶ X to equal f^{(1)} outside of S. On these sectors, f^{(2)} is defined by induction. Let n_{i} denote the integer determined by

f^{(1)} : ∂(S(\frac{i}{q + 1})) ⟶ ∂(S̃(n_{i} + \frac{i + p}{q + 1})).

Choose for 1 ≤ i ≤ q a diffeomorphism

f_{0,i}^{(2)} : T_{0}(\frac{i}{q + 1}) ⟶ T̃_{0}(n_{i} + \frac{i + p}{q + 1})

so that f^{(1)} and f_{0,i}^{(2)} determine the same tangent map on the boundary of the sectors.

Define f_{k,i}^{(2)} : T_{k}(\frac{i}{q+1}) ⟶ T̃_{k}(n_{i} + \frac{i+p}{q+1}) by induction as the map satisfying

Π̃_{n_{i}+ \frac{i+p}{q+1}} ∘ f_{k,i}^{(2)} = f_{k−1,i}^{(2)} ∘ Π_{\frac{i}{q+1}} .

Finally, extend f^{(2)} : X′ ⟶ X as a covering transformation so that it is compatible with the projection. The map f_{λ}^{(2)} : X′_{λ} ⟶ X_{λ} is defined on X′_{λ} \ K_{λ}^{E} by

X′  ⟶^{f^{(2)}} X
(ψ_{λ}∘exp)^{E} ↓  ↓^{(ψ_{λ}∘exp)^{E}}
X′_{λ} \ K_{λ}^{E} ⟶^{f_{λ}^{(2)}} X_{λ} \ K_{λ}^{E}

and f_{λ}^{(2)}|_{K_{λ}^{E}} = f_{λ}^{(1)}|_{K_{λ}^{E}}.

The map f^{(2)} : X′ ⟶ X is a diffeomorphism. The map f_{λ}^{(2)} : X′_{λ} ⟶ X_{λ} is a degree 2-ramified covering.

Lemma 5.8 The map f^{(2)} is quasi-conformal. Hence f_{λ}^{(2)} is quasi-regular.

Proof : As in Lemma 5.1, the map f^{(2)} : X′ ⟶ X is quasi-conformal, since f_{0,i}^{(2)} are diffeomorphisms on compact sets and f_{k,i}^{(2)} is obtained from f_{0,i}^{(2)} by composition with holomorphic maps.

q.e.d.

We change the complex structure (as we did in Sect. 5.2) into an almost complex structure σ, which is quasi-conformally equivalent to the standard structure, σ_{0}, and such that it is invariant under the map f^{(2)}. Also, σ coincides with σ_{0} outside of S and its preimages. Here, as in the quadratic case, orbits pass through these sectors at most once, since after one iteration, a point in S is mapped to a point in S̃ and do not leave S̃ as long as the map is defined.

By construction σ is invariant under f^{(2)}. Define σ_{λ} on X_{λ} \ K_{λ}^{E} as the pullback of σ by the map ((ψ_{λ} ∘ exp)^{E})^{−1} and set σ_{λ} = σ_{0} on K_{λ}^{E}. (Compare with Lemma 5.3 on page 38).

We apply the Measurable Riemann Mapping Theorem to obtain the unique quasiconformal homeomorphism (in this case also a hybrid equivalence (see remark 5.2))

φ_{λ} :X^{∘}_{λ}⟶ 𝔻,

integrating σ_{λ} and satisfying φ_{λ}(ω_{λ}) = 0 and φ_{λ}(0) ∈ ℝ_{+}. Then, let D′_{λ} = φ_{λ}(X^{∘}′_{λ}) and define

f_{λ}^{(3)} = φ_{λ} ∘ f_{λ}^{(2)} ∘ φ_{λ}^{−1} : D′_{λ} ⟶ 𝔻.

The map f_{λ}^{(3)} is a polynomial-like map of degree two, which has one critical point at φ_{λ}(−q/(q +1)). We now apply the Straightening Theorem to obtain a uniquely determined polynomial Q_{c(λ)} and a hybrid equivalence χ_{λ} conjugating f_{λ}^{(3)} to Q_{c(λ)}. Hence we define

ξ_{p/q}(λ) = c(λ).

Remarks 5.9 By the same arguments as in remarks 5.5 on page 40 the value of c does not depend on the choices made throughout the construction, i.e. the choice of the boundaries γ and γ′, the slope s and the diffeomorphisms f_{0,i}^{(2)}. We may also assume χ_{λ} to be defined on all φ_{λ}(X_{λ}).

The following proposition is the analogue of Theorem G for the inverse function.

Proposition 5.10 Suppose λ ∈ L_{q,0} and c = ξ_{p/q}(λ) ∈ M_{p/q}. Recall that X_{λ} is a neighborhood of K_{λ}^{E} bounded by the extended equipotential γ_{λ} and that W_{c} is a neighborhood of K_{c}, the filled level set of a chosen equipotential of G_{c}. There exists a homeomorphism

H_{λ} : X_{λ} ⟶ W_{c}

which satisfies \overline{∂}H_{λ} = 0 on K_{λ}^{E} and conjugates f_{λ}^{(2)} to Q_{c} and such that H_{λ} maps relevant sectors and rays for P_{λ} to their counterparts for Q_{c}, i.e. H_{λ} maps S̃_{λ}(\frac{i}{q+1}), R_{λ}(\frac{i}{q+1}), and R̃_{λ}(\frac{i}{q+1}) to S_{c}(θ̃^{i−1}), R_{c}(θ^{i−1}), and R_{c}(θ̃^{i−1}) respectively.

Moreover, H_{λ}|_{K_{λ}^{E}} is uniquely determined and it conjugates f_{λ}^{(1)} to Q_{c}.

The proof is analogous to the proof of Theorem G.

Substituting c by λ, remarks 5.7 hold for ξ_{p/q}.

Proposition 5.11 1. ϕ_{p/q} ∘ ξ_{p/q} = Id_{L_{q,0}}

2. ξ_{p/q} ∘ ϕ_{p/q} = Id_{M_{p/q}}

Hence ϕ_{p/q} is bijective.

Proof : To prove the first statement start with λ ∈ L_{q,0} and, following the construction above, obtain a polynomial Q_{c} and the map H_{λ} given by Prop. 5.10, where c = ξ_{p/q}(λ). We now apply the construction in section 5.2 on page 32 to Q_{c} choosing

γ_{c} = H_{λ}(γ_{λ}) ∩ ℂ_{c}^{T}
γ′_{c} = H_{λ}(P_{λ}^{−1}(γ_{λ})) ∩ ℂ_{c}^{T}
f_{c,0,i}^{(2)} = H_{λ} ∘ f_{λ,0,i}^{(2)} ∘ H_{λ}^{−1}

Let λ′ = ϕ_{p/q}(c) which, by the remarks 5.5 on page 40, does not depend on these choices. Let H_{c} : X_{c} ⟶ W_{λ′} be given by Theorem G. It is easy to check that H_{c} ∘ H_{λ}|_{W_{λ}^{T}} is a hybrid equivalence between P_{λ} and P_{λ′}. Hence λ = λ′ = ϕ_{p/q}(ξ_{p/q}(λ)) by Proposition 3.3.

To prove the second statement choose c ∈ M_{p/q} and set λ = ϕ_{p/q}(c) and c′ = ξ_{p/q}(λ). We must show that Q_{c} ∼_{hb} Q_{c′} and therefore c = c′.

Let H_{c} be given by Theorem G. Besides the properties of H_{c} stated in the theorem and since H_{c} conjugates f_{c}^{(2)} to P_{λ} we have that

• the curves (f_{c}^{(2)})^{−n}(γ′_{c}) are mapped to equipotentials of P_{λ} of potential η′(q+1)^{(n+1)/q},

• the rays landing at α′_{c} are mapped accordingly to the rays landing at −q. Moreover, if potential ρ on R_{c}(θ̃) corresponds to potential ρ′ on R_{λ}(0), then for all i ∈ ℝ_{+}, potential ρ2^{−i} in R_{c}(θ̃) corresponds to potential ρ′(q + 1)^{−i/q} on R_{λ}(0).

We provide the idea of the construction, leaving the details to the reader.

To define the domain of f_{λ}^{(2)} : X′_{λ} ⟶ X_{λ} choose

γ_{λ} ∩ V_{λ}^{0...q−1} = H_{c}(γ_{c})
γ′_{λ} ∩ V_{λ}^{0...q−1} = H_{c}(Q_{c}^{−1}(γ_{c})),

and complete the curves in Ṽ_{λ}^{i} as usual.

We now extend H_{c} : X_{c} → W_{λ} to H_{c}^{E} : W_{c} → X_{λ} as

H_{c}^{E} = \begin{cases}τ_{λ}^{ip} ∘ P_{λ}|_{V_{λ}^{ip}}^{−1} ∘ H_{c} ∘ Q_{c}^{q−i} & if z ∈ Ṽ_{c}^{ip} \\ H_{c} & otherwise \end{cases}

One can check that H_{c}^{E} is well defined along the rays landing at 0. It satisfies \overline{∂}H_{c}^{E} = 0 on K_{c}.

Finally, we define f_{λ}^{(2)} on the sectors where the first return map is modified so that

f_{λ}^{(2)} = H_{c}^{E} ∘ Q_{c} ∘ H_{c}^{−1}

and f_{λ}^{(1)} elsewhere. One can check that this map is a valid choice and that the following diagram

W_{c,1} ⟶^{Q_{c}} W_{c}
H_{c}^{E} ↓  ↓^{H_{c}^{E}}
X′_{λ} ⟶^{f_{λ}^{(2)}} X_{λ}

commutes.

From the above, H_{c}^{E} is a hybrid equivalence between Q_{c} and f_{λ}^{(2)}. Since c′ does not depend on the choices, we have that f_{λ}^{(2)} is hybrid equivalent to Q_{c′}. Hence Q_{c} and Q_{c′} are hybrid equivalent.

q.e.d.

5.4 Analyticity of ϕ_{p/q} in the interior of M_{p/q}

Let Ω_{M} be a hyperbolic component of M_{p/q}. Then ϕ_{p/q} maps Ω_{M} to Ω_{L}, a hyperbolic component of L_{q,0} (as defined in section 3). By remark 5.7 on page 42, the multiplier of the corresponding attracting cycles is preserved. Let ρ_{Ω_{M}} : 𝔻 → Ω_{M} (resp. ρ_{Ω_{L}} : 𝔻 → Ω_{L}) be the multiplier function that parametrizes Ω_{M} (resp. Ω_{L}). Then, by the observation above

ϕ_{p/q} |_{Ω_{M}} = ρ_{Ω_{L}} ∘ ρ_{Ω_{M}}^{−1} and ξ_{p/q} |_{Ω_{L}}= ρ_{Ω_{M}} ∘ ρ_{Ω_{L}}^{−1}  (9)

and hence ϕ_{p/q} is holomorphic on all hyperbolic components.

If the interior of M were known to equal the union of its hyperbolic components we would be done. However, this being still a conjecture implies that there might be non-hyperbolic components in the interior of M (also called queer components). Such components might exist similarly for L_{q}. In both cases, the filled Julia sets of polynomials that belong to nonhyperbolic components have empty interior due to the classification of Fatou components.

Let Ω̃_{M} be a non-hyperbolic component of M_{p/q}. In order to prove analyticity of ϕ_{p/q} in Ω̃_{M} we will use a parametrization similar to the parametrization of a hyperbolic component given by the multiplier map. In what follows we describe how to obtain such a parametrization.

Definition Let X be a Riemann surface. A line field supported on E ⊂ X is an L^{∞} Beltrami form μ supported on E with ∥μ∥_{∞} = 1.

Proposition 5.12 A parameter c belongs to a non-hyperbolic component of M if and only if J_{c} has positive measure and carries an invariant line field.

(For the proof see [MSS] and in particular p.61 in [Mc])

Remark 5.13 The same proposition is true for a polynomial P_{q,λ}, with λ in a nonhyperbolic component of the interior of L_{q}. •

Proposition 5.14 Let Ω̃_{M} (resp. Ω̃_{L}) be a non-hyperbolic component of M (resp. L_{q}), and let c, c′ ∈ Ω̃_{M} (resp. λ, λ′ ∈ Ω̃_{L}). Then, there exists a unique co-hybrid equivalence (see Sect.2.2) ψ_{cc′} : ℂ → ℂ (resp. ψ_{λλ′} : ℂ → ℂ) between Q_{c} and Q_{c′} (resp. P_{λ} and P_{λ′}) such that \frac{ψ_{cc′}(z)}{z} → 1 (resp. \frac{ψ_{λλ′}(z)}{z} → (\frac{λ}{λ′})^{\frac{1}{q}}) as z → ∞. Moreover, if c_{i} ∈ Ω̃_{M} (resp. λ_{i} ∈ Ω̃_{L}), i = 1, 2, and μ_{i} = \frac{\overline{∂}ψ_{cc_{i}}}{∂ψ_{cc_{i}}} (resp. μ_{i} = \frac{\overline{∂}ψ_{λλ_{i}}}{∂ψ_{λλ_{i}}}), i = 1, 2, then there exists t ∈ ℂ^{∗} such that μ_{1} = tμ_{2}.

Proof : We give the idea of the proof in the case of Ω̃_{L} (compare with [Mc]). For the first part, set ψ_{λλ′} = ψ_{λ′} ∘ ψ_{λ}^{−1} : ℂ \ K_{λ} ⟶ ℂ \ K_{λ′}. By the λ-lemma, ψ_{λλ′} extends to a quasi-conformal map from ℂ to ℂ conjugating P_{λ} to P_{λ′}. The uniqueness follows from the uniqueness of ψ_{λ} and ψ_{λ′} as defined in Sect. 4.2. Finally, observe that

\frac{ψ_{λλ′}}{z} = \frac{g_{ν′}g_{ν}^{−1}(z)}{z} → \frac{ν}{ν′} = (\frac{λ}{λ′})^{\frac{1}{q}} as z → ∞.

To show the uniqueness of the Beltrami forms up to multiplication by a constant observe that if that were not the case we could construct from the two parameter family t_{1}μ_{1}+t_{2}μ_{2} of Beltrami forms, an injective map (t_{1}, t_{2}) ↦ λ(t_{1}, t_{2}) giving rise to a two complex parameter family of polynomials of the form P_{q,λ}, which would be a contradiction.

q.e.d.

Corollary 5.15 For c ∈ Ω̃_{M} (resp. λ ∈ Ω̃_{L}), an invariant line field on K_{c} (resp. K_{λ}) is unique up to rotation.

Proposition 5.16 Let c ∈ Ω̃_{M} and let μ_{c} denote an invariant line field of Q_{c}. The Beltrami forms (tμ_{c})_{t∈𝔻} induce a conformal equivalence

ρ_{c,μ_{c}} : 𝔻 ⟶ Ω̃_{M}.

Similarly, if λ ∈ Ω̃_{L} and μ_{λ} denotes an invariant line field of P_{q,λ}, the Beltrami forms (tμ_{λ})_{t∈𝔻} induce a conformal equivalence

ρ_{λ,μ_{λ}} : 𝔻 ⟶ Ω̃_{L}.

Proof : We give the proof for the case of Ω̃_{L} (compare with [Mc]). For the proof in the case of Ω̃_{M} just replace λ and P by c and Q respectively.

For t ∈ 𝔻 consider the Beltrami form tμ_{λ}. Then ∥tμ_{λ}∥_{∞} = |t| < 1 and tμ_{λ} is invariant under P_{λ}. Apply the Measurable Riemann Mapping Theorem (dependent on parameters) to obtain a quasi-conformal homeomorphism φ_{λ}^{t} that integrates tμ_{λ} and such that it fixes −q, 0 and ∞. The map φ_{λ}^{t} ∘ P_{λ} ∘ (φ_{λ}^{t})^{−1} is holomorphic and a degree q + 1 polynomial of the form P_{λ(t)}. Define ρ_{λ,μ_{λ}}(t) = λ(t).

To show that λ(t) is holomorphic note that the free critical point (resp. free critical value) of P_{λ} must be mapped under φ_{λ}^{t} to the free critical point (resp. free critical value) of P_{λ(t)}. Hence

λ(t) = −φ_{λ}^{t}(−λ(\frac{q}{q + 1})^{q+1})(\frac{q + 1}{q})^{q+1}.

Since tμ_{λ} varies holomorphically with t, φ_{λ}^{t} varies holomorphically with t and therefore so does λ(t).

To show that λ(t) is injective, suppose that λ(t_{1}) = λ(t_{2}). Then

φ_{λ}^{t_{1}} ∘ P_{λ} ∘ (φ_{λ}^{t_{1}})^{−1} = φ_{λ}^{t_{2}} ∘ P_{λ} ∘ (φ_{λ}^{t_{2}})^{−1}

i.e. P_{λ} = (φ_{λ}^{t_{2}})^{−1} ∘ φ_{λ}^{t_{1}} ∘ P_{λ} ∘ (φ_{λ}^{t_{1}})^{−1} ∘ φ_{λ}^{t_{2}}.

Hence φ = (φ_{λ}^{t_{2}})^{−1} ∘ φ_{λ}^{t_{1}} is a co-hybrid equivalence between P_{λ} and itself. Let ψ_{λ} be the Böttcher coordinate on ℂ \ K_{λ}. Then ψ_{λ}^{−1} ∘ φ ∘ ψ_{λ} : ℂ \ \overline{𝔻} ⟶ ℂ \ \overline{𝔻} is holomorphic and conjugates z ↦ z^{q+1} to itself. Moreover, φ maps the cycle of rays landing at the repelling fixed point 0 onto themselves. Hence φ must be the identity. Since ψ_{λ} is bijective, this implies φ_{λ}^{t_{1}} = φ_{λ}^{t_{2}} on ℂ \ K_{λ}.

Since K_{λ} = ∂(ℂ \ K_{λ}) and φ_{λ}^{t_{1}} and φ_{λ}^{t_{2}} are continuous, it follows that φ_{λ}^{t_{1}} = φ_{λ}^{t_{2}} on K_{λ} and therefore on all of ℂ.

Finally, \frac{\overline{∂}φ_{λ}^{t_{i}}}{∂φ_{λ}^{t_{i}}} = t_{i}μ_{λ} for i = 1, 2, so t_{1}μ_{λ} = t_{2}μ_{λ}. Since μ_{λ} ≠ 0 on a set of positive

measure we have that t_{1} = t_{2}.

At this point we have obtained a holomorphic injective map from the unit disc to a neighborhood of λ in Ω̃_{L}. To show that this neighborhood covers the whole component, pick a point λ′ ∈ Ω̃_{L}. By Prop. 5.14 there exists a co-hybrid equivalence ψ_{λλ′} between P_{λ} and P_{λ′} with μ′ := \frac{\overline{∂}ψ_{λλ′}}{∂ψ_{λλ′}}. Moreover, since φ_{λ}^{t} is a co-hybrid equivalence between P_{λ} and P_{λ(t)} for any t ∈ 𝔻, we have that μ′ = t^{∗}tμ_{λ} for some t^{∗} ∈ ℂ. Hence ψ_{λλ′} ∘ ψ_{λ}^{t^{∗}t} is a conformal equivalence between P_{λ′} and P_{λ(t^{∗}t)} and therefore λ′ = λ(t^{∗}t).

q.e.d.

Definition/Lemma 5.17 Let c ∈ Ω̃_{M}, λ = ϕ_{p/q}(c) and H_{c} be a quasi-conformal homeomorphism as in Theorem G, uniquely defined on K_{c}^{T}. Given μ_{c} an invariant line field on K_{c}, let μ_{c}^{T} be the restriction of μ_{c} to ℂ_{c}^{T}. Then the Beltrami form

μ_{λ} = \begin{cases}(H_{c}^{−1})^{∗}μ_{c}^{T} & on K_{λ} \\ 0 & on ℂ \ K_{λ}. \end{cases}

defines an invariant line field in K_{λ} which we call the induced line field on K_{λ}.

Proof : Note that μ_{λ} is well defined since H_{c}|_{K_{c}^{T}} is unique. The fact that ∥μ_{λ}∥_{∞} = 1 follows from \overline{∂}H_{c} = 0 on K_{c}^{T}.

Let E ⊂ K_{c}^{T} be the set of positive measure where μ_{c}^{T} ≠ 0. Since quasi-conformal mappings are absolutely continuous with respect to the Lebesgue measure, we have that H_{c}(E) ⊂ K_{λ} has positive measure. Hence μ_{λ} ≠ 0 on a set of positive measure.

The invariance of μ_{λ} by P_{λ} follows from the fact that H_{c} conjugates f_{c}^{(1)} to P_{λ}. By Prop. 5.12, λ belongs to a non-hyperbolic component of L_{q,0}.

q.e.d.

Theorem 5.18 Let Ω̃_{M} be a non-hyperbolic component of M_{p/q}. Then, Ω̃_{L} := ϕ_{p/q}(Ω̃_{M}) is a non-hyperbolic component of L_{q,0}.

Moreover, if we fix c ∈ Ω̃_{M}, λ = ϕ_{p/q}(c) and μ_{c} an invariant line field on K_{c}, then

ϕ_{p/q}|_{Ω_{M}} = ρ_{λ,μ_{λ}} ∘ ρ_{c,μ_{c}}^{−1}

where μ_{λ} denotes the induced line field on K_{λ}; hence ϕ_{p/q} is holomorphic on all nonhyperbolic components of M.

The rest of the section is dedicated to prove the theorem above.

Pick c ∈ Ω̃_{M} and let μ_{c}, ρ_{c} and φ_{c}^{t} be as in proposition 5.16. To prove that ϕ_{p/q} is holomorphic in Ω̃_{M} we will use notation from the construction of this map in section 5.2 on page 32.

Let (φ_{c}^{t})^{T} = φ_{c}^{t}|_{ℂ_{c}^{T}}.

Lemma 5.19 The diagram

X′_{c} ⟶^{f_{c}^{(2)}} X_{c}
(φ_{c}^{t})^{T} ↓  ↓^{(φ_{c}^{t})^{T}}
X′_{c(t)} ⟶^{f_{c(t)}^{(2)}} X_{c(t)}

commutes and \frac{\overline{∂}(φ_{c}^{t})^{T}}{∂(φ_{c}^{t})^{T}} = tμ_{c}^{T}.

Proof : Let ψ_{c} and ψ_{c(t)} be as above. We have the following diagram

ℂ \ \overline{𝔻} ⟶^{z↦z^{2}} ℂ \ \overline{𝔻}
ψ_{c}↓  ↓^{ψ_{c}}
ℂ \ K_{c} ⟶^{Q_{c}} ℂ \ K_{c}
φ_{c}^{t}↓  ↓^{φ_{c}^{t}}
ℂ \ K_{c(t)} ⟶^{Q_{c(t)}} ℂ \ K_{c(t)}
ψ_{c(t)}↑  ↑^{ψ_{c(t)}}
ℂ \ \overline{𝔻} ⟶^{z↦z^{2}} ℂ \ \overline{𝔻}

By the same argument as before ψ_{c(t)}^{−1} ∘ φ_{c}^{t} ∘ ψ_{c} = Id|_{ℂ\\overline{𝔻}}. It follows that φ_{c}^{t} maps rays to rays and equipotentials to equipotentials. Hence (φ_{c}^{t})^{T} (ℂ_{c}^{T}) = ℂ_{c(t)}^{T}.

Let

(ℂ \ \overline{𝔻})^{T} = \{exp(ρ + 2πit) ∈ ℂ \ \overline{𝔻} | ρ + 2πit ∈ ℍ^{T} \}

and ψ_{c}^{T} = ψ_{c}|_{(ℂ\\overline{𝔻})^{T}}. Then, f^{(2)} : X′ → X (X′ ⊂ X ⊂ ℍ^{T}) induces a map f̃^{(2)} : X̃′ → X̃ (X̃′ ⊂ X̃ ⊂ (ℂ \ \overline{𝔻})^{T}) such that the following diagram commutes

X̃′ \underrightarrow{_{f^{(2)}}} X̃
ψ_{c}^{T} ↓  ↓^{ψ_{c}^{T}}
X′_{c} ⟶^{f_{c}^{(2)}} X_{c}

and the same is true replacing c by c(t). Then,

(φ_{c}^{t})^{T} (f_{c}^{(2)}(ψ_{c}^{T} (z))) = (φ_{c}^{t})^{T} (ψ_{c}^{T} (f̃^{(2)}(z)))
= ψ_{c(t)}^{T}(f̃^{(2)}(z))
= f_{c(t)}^{(2)}(ψ_{c(t)}^{T}(z))
= f_{c(t)}^{(2)}((φ_{c}^{t})^{T} (ψ_{c(t)}^{T}(z)))

which proves that the diagram in the lemma commutes.

Since (φ_{c}^{t})^{T} is a restriction of φ_{c}^{t} we have that \frac{\overline{∂}(φ_{c}^{t})^{T}}{∂(φ_{c}^{t})^{T}} = tμ_{c}^{T}. q.e.d.

Lemma 5.20 The map H := H_{c(t)} ∘ (φ_{c}^{t})^{T} ∘ H_{c}^{−1} conjugates P_{λ} to P_{ϕ_{p/q}(c(t))}. Moreover,

\frac{\overline{∂}H}{∂H} = tμ_{λ} on K_{λ}.

Proof : The conjugacy follows from the construction of ϕ_{p/q} together with lemma 5.19. Now consider the Beltrami form tμ_{λ} on K_{λ}. By H_{c}, tμ_{λ} is transported to tμ_{c}^{T} since \overline{∂}H_{c} = 0 on K_{c}. The map (φ_{c}^{t})^{T} integrates tμ_{c}^{T} which means that tμ_{c}^{T} is transported to the standard complex structure on K_{c(t)}^{T}. Now, \overline{∂}H_{c} = 0 on K_{c(t)} so in summary, tμ_{λ} is transported to the standard complex structure by H.

q.e.d.

We have proven that the following diagram commutes

N(K_{ϕ_{p/q}(c(t))}) ⟶^{P_{ϕ_{p/q}(c(t))}} N(K_{ϕ_{p/q}(c(t))})
H↑  ↑^{H}
N(K_{λ})  ⟶^{P_{λ}}  N(K_{λ})
φ_{λ}^{t}↓  ↓^{φ_{λ}^{t}}
N(K_{λ(t)})  ⟶^{P_{λ(t)}}  N(K_{λ(t)})

Moreover, \overline{∂}(φ_{λ}^{t} ∘ H^{−1}) = 0 on K_{ϕ_{p/q}(c(t))} since both maps H and φ_{λ}^{t} integrate tμ_{λ}. Hence, φ_{λ}^{t}∘H^{−1} is a hybrid equivalence between P_{λ(t)} and P_{ϕ_{p/q}(c(t))} and therefore λ(t) = ϕ_{p/q}(c(t)) by Cor. 3.3.

q.e.d. Thm. 5.18

Remark 5.21 One could state Theorem 5.18 in terms of ξ_{p/q} concluding that this map is also holomorphic. One should mimic the proof above considering the “extended” versions of φ_{λ}^{t} and μ_{λ} as opposed to the “truncated” versions of φ_{c}^{t} and μ_{c}. •

5.5 Continuity of ϕ_{p/q} at points on the boundary of M_{p/q}

Assume that c is in the boundary of the p/q-limb. The following will be a key lemma:

Lemma 5.22 Suppose λ ∈ ∂L_{q}, λ′ ∈ ℂ and P_{λ} ∼_{qc} P_{λ′}. Then λ = λ′.

Proof : Let φ be the quasi-conformal homeomorphism that conjugates P_{λ} to P_{λ′}, i.e.

P_{λ′} = φ ∘ P_{λ} ∘ φ^{−1}

Set μ = \frac{\overline{∂}φ}{∂φ}. Since φ is quasi-conformal, we have that ∥μ∥_{∞} < 1.

Define a new Beltrami form μ̃ by

μ̃ = \begin{cases}μ & on K_{λ} \\ 0 & on ℂ \ K_{λ} \end{cases}

Then k = ∥ μ̃ ∥_{∞} ≤ ∥ μ ∥_{∞} < 1.

Consider the family of Beltrami forms μ_{t} = tμ̃, for |t| < 1/k. Then,

∥ μ_{t} ∥_{∞} = |t| ∥ μ̃ ∥_{∞} = |t| k < 1.

We then apply the Measurable Riemann Mapping Theorem (dependent on parameters) to obtain the unique integrating maps φ_{t} : ℂ ⟶ ℂ such that \frac{\overline{∂}φ_{t}}{∂φ_{t}} = μ_{t}, and such that they fix 0, ∞ and −q. Since μ_{t} depends analytically on t, the maps φ_{t} also depend analytically on t.

For each t, let P_{t} = φ_{t} ∘ P_{λ} ∘ φ_{t}^{−1} which is a polynomial. Because of the choices made for φ_{t}, it is easy to check that P_{t} must be of type E. Hence, P_{t} = P_{λ(t)} = P_{q,λ(t)} and φ_{t}(ω) = ω, where ω = −q/(q + 1) is the simple critical point. Also, the orbit of ω by P_{λ(t)} must be bounded, so λ(t) ∈ L_{q}.

By the same argument as in the proof of Prop. 5.16, λ(t) is an analytic function of t.

Now note that μ_{0} = 0, so φ_{0} = Id and hence λ(0) = λ. Since λ(t) is analytic then, by the open mapping principle, it is either open or constant. If it is open, a neighborhood of t = 0 must be mapped to a neighborhood of λ, which is a contradiction since λ ∈ ∂L_{q} and λ(t) belongs to L_{q} for all t. Therefore λ(t) is constant and since λ(0) = λ it follows that λ(t) ≡ λ for all t.

In particular λ(1) = λ. But for t = 1 we have μ_{1} = μ̃. Hence, we have

ℂ ⟶^{P_{λ}} ℂ
φ_{1}↑  ↑^{φ_{1}}
ℂ ⟶^{P_{λ}} ℂ
φ↓  ↓^{φ}
ℂ ⟶^{P_{λ′}} ℂ

and φ ∘ φ_{1}^{−1} is another quasi-conformal homeomorphism that conjugates P_{λ} to P_{λ′}. In fact, we claim that it is a hybrid equivalence since it pulls back the standard complex structure on K_{λ′} to the standard complex structure on K_{λ}. It follows that P_{λ} ∼_{hb} P_{λ′} and therefore by Prop. 3.3, λ = λ′.

q.e.d.

Given any sequence c_{n} → c, let λ_{n} = ϕ_{p/q}(c_{n}) and λ = ϕ_{p/q}(c). To prove continuity at c we must show that lim_{n→∞} λ_{n} = λ.

We claim that it suffices to pick an arbitrary convergent subsequence λ_{n_{k}} → λ̃, and show that λ̃ = λ. Indeed, since λ_{n_{k}} is arbitrary, it follows that all convergent subsequences tend to λ = ϕ_{p/q}(c). Since L_{q,0} ∪ {1} is compact, it follows that lim_{n→∞} λ_{n} = λ.

So, from now on we assume that we have picked a sequence c_{n} → c ∈ ∂M_{p/q}. Let φ_{n}, f_{n}^{(3)}, χ_{n}, etc. be as in section 5.2. To make the notation easier we denote the convergent subsequence again by λ_{n} → λ̃. We want to show that λ̃ = λ.

Lemma 5.23 The polynomials P_{λ} and P_{λ} are quasi-conformally conjugate.

Proof : The mappings φ_{n} are all quasi-conformal homeomorphisms and they have a uniform bounded dilatation ratio, since a new complex structure was constructed once and for all in the right half plane and then pulled back by (ψ_{λ} ∘ exp)^{T}. Let K be this uniform bound. Then, {φ_{n}} is a K-quasi-conformal family. Since the space of K-quasi-conformal mappings on a compact set is compact with respect to uniform convergence, we have that the {φ_{n}} form an equi-continuous family. Therefore, by Arzela-Ascoli’s theorem, we can find a convergent subsequence φ_{n_{k}} → φ^{∗}. The limit map is a K-quasi-conformal homeomorphism. The map

f_{∗}^{(3)} = φ^{∗} ∘ f_{c}^{(2)} ∘ (φ^{∗})^{−1}.

is analytic and

f_{∗}^{(3)} ∼^{φ^{∗}}_{hb} f^{(2)} ∼^{φ_{c}}_{hb} f_{c}^{(3)}.

Hence f_{c}^{(3)} ∼_{qc} f_{∗}^{(3)}.

We now apply the lemma in p. 313 in [DH3] to f_{∗}^{(3)} to obtain a subsequence of hybrid equivalences {χ_{n}} (abusing notation again) that converges to a quasi-conformal homeomorphism χ^{∗} and a polynomial P^{∗} such that P_{λ_{n}} → P^{∗} and

P_{∗} = χ^{∗} ∘ f_{∗}^{(3)} ∘ (χ^{∗})^{−1}.

Then

P^{∗} ∼^{χ^{∗}}_{hb} f^{(3)} ∼^{χ_{c}}_{hb} P_{λ}.

so it follows that P_{λ} ∼_{qc} P^{∗}.

Finally, note that P^{∗} must be of the form P_{λ^{∗}} (maybe after composing with an affine conjugacy). Hence, P_{λ_{n}} → P^{∗} implies that λ_{n} → λ^{∗}. It follows that λ^{∗} = λ̃ and therefore we have proved P_{λ} ∼_{qc} P_{λ}.

q.e.d.

We now give the conclusion of the proof.

Suppose the sequence c_{n} → c was such that all the c_{n}’s are Misiurewicz points. Then it is clear the λ_{n}’s are also Misiurewicz points and hence λ_{n} ∈ ∂L_{q}. Since the boundary is compact we have that any accumulation point of {λ_{n}} must be on the boundary. So λ̃ ∈ ∂L_{q}.

By Lemma 5.23, P_{λ} ∼_{qc} P_{λ}. This fact, together with λ̃ ∈ ∂L_{q} allows as to apply Lemma 5.22 to obtain λ̃ = λ.

Now suppose the sequence c_{n} → c was an arbitrary sequence, so that there is no assumption on the c_{n}’s. Choose also another sequence c′_{n} → c formed by Misiurewicz points, and let λ′ be an accumulation point of {λ′_{n}}. Then, by the above we have that λ′ = λ and hence λ ∈ ∂L_{q}. Still, by lemma 5.23 we have P_{λ} ∼_{qc} P_{λ}. This fact, together with λ ∈ ∂L_{q}, allows us to use again Lemma 5.22 to conclude λ = λ̃.

This concludes the proof of continuity at points on the boundary.

This concludes the proof of Theorem F.

Remark 5.24 It follows that (9) is satisfied on the closure \overline{Ω_{M}}, so if c has internal argument t then λ = ϕ_{p/q}(c) has also internal argument t. In particular, roots of hyperbolic components are mapped to roots of hyperbolic components. •

5.6 Proof of theorem A

At this point, for every p/q, we have constructed a homeomorphism

ϕ_{p/q} : M_{p/q} ⟶ L_{q,0}

which is analytic in the interior of M_{p/q}.

Now, for any p/q and p′/q, define the map Φ_{pp′}^{q} to be

Φ_{pp′}^{q} : M_{p/q} ⟶^{ϕ_{p/q}} L_{q,0} ⟶^{ϕ_{p′/q}^{−1}} M_{p′/q}

.

Thus, Φ_{pp′}^{q} is a homeomorphism between M_{p/q} and M_{p′/q} which by Theorem 5.18 is holomorphic in the interior of the limbs.

For a given c ∈ M_{p/q}, the second part of Theorem A claims the existence of a homeomorphism Φ̂_{c} between a neighborhood of K_{c} and a neighborhood of K_{c′} where c′ = Φ_{pp′}^{q}(c).

Let λ = ϕ_{p/q}(c) = ϕ_{p′/q}(c). Let H_{c} : X_{c} ⟶ W_{λ} and H_{c′} : X_{c′} ⟶ W_{λ} be quasiconformal homeomorphisms as in Theorem G. Recall that these maps conjugate the first return maps to the polynomial P_{λ} outside of the sectors and send the sector S_{c}(θ̃) (resp S_{c′}(θ̃)) around the identified ray (R_{c}(θ̃), resp. R_{c′}(θ̃)) to the sector S_{λ}(0) around R_{λ}(0). Define,

Φ̂_{c}^{T} : X_{c} \ S_{c}(θ̃) ⟶^{H_{c}} W_{λ} \ S_{λ}(0) ⟶^{H_{c′}^{−1}} X_{c′} \ S_{c′}(θ̃).

Finally, extend Φ̂_{c}^{T} to W_{c} - the filled level set of the chosen potential - by defining

Φ̂_{c}(z) = \begin{cases}Φ̂_{c}^{T} (z) & if defined \\ −Φ̂_{c}^{T} (z) & otherwise \end{cases}

Lemma 5.25 The map Φ̂_{c} is well defined.

Proof : We must check that along the boundaries of S_{c}(θ̃^{0}) and S_{c}(θ̃^{q−1}) in V_{c}^{0} we have

Φ̂_{c}^{T} (−z) = −Φ̂_{c}^{T} (z).

We will actually prove it for all points in V_{c}^{0} not in the interior of the sectors. We know that in this region Φ̂_{c}^{T} conjugates Q_{c}^{q} to Q_{c′}^{q}. Hence

Φ̂_{c}^{T} (Q_{c}^{q}(z)) = Q_{c′}^{q}(Φ̂_{c}^{T} (z))

and the same holds for −z since −z ∈ V_{c}^{0}. But Q_{c}^{q}(z) = Q_{c}^{q}(−z) so

Q_{c′}^{q}(Φ̂_{c}^{T} (z)) = Q_{c′}^{q}(Φ̂_{c}^{T} (−z)).

Since Q_{c}^{q} is two-to-one on V_{c}^{0} the conclusion follows.

q.e.d.

This concludes the proof of Theorem A.

Remark 5.26 The map Φ̂_{c} is a quasi-conformal homeomorphism satisfying \overline{∂}Φ̂_{c} = 0 on K_{c}. This map does not conjugate Q_{c} and Q_{c′} but it conjugates the first return map on K_{c}^{T} to the first return map on K_{c′}^{T}. •

5.7 Compatibility with Tuning

The Mandelbrot set contains copies of itself. For each center c_{0} of a hyperbolic component there is associated a copy of M obtained by tuning (see [D]):

\begin{matrix}M & ⟶ & M_{c_{0}} = c_{0} ⊥ M \\ c & ⟶ & c_{0} ⊥ c \end{matrix}

The tuning map is a homeomorphism mapping centers of hyperbolic components to centers of hyperbolic components and Misiurewicz points to Misiurewicz points; it is analytic in the interior of M and ∂M_{c_{0}} ⊂ ∂M. Moreover, Misiurewicz points are dense in ∂M (see [Br]), hence dense in ∂M_{c_{0}}.

Proposition 5.27 Let c_{0} be a center of a hyperbolic component in M_{p/q}. The homeomorphism Φ_{pp′}^{q} is compatible with tuning, i.e.

Φ_{pp′}^{q}(c_{0} ⊥ c) = Φ_{pp′}^{q}(c_{0}) ⊥ c  (10)

for all c ∈ M.

Figure 31: Left: the Hubbard tree for c_{0} ⊥ (−2), where c_{0} is the center of the period 6 hyperbolic component in M_{1/5} (as in Fig. 3 left). Right: the Hubbard tree for ϕ_{12}^{5}(c_{0}) ⊥ (−2), where ϕ_{12}^{5}(c_{0}) is the center of the period 8 hyperbolic component (as in Fig. 3 right). Compare also with Fig. 4.

Proof : The parameter c′_{0} = Φ_{pp′}^{q}(c_{0}) is the center of a hyperbolic component in M_{p′/q}. Following the surgery constructions it is very easy to see that eq. (10) is satisfied for any Misiurewicz point c ∈ M (compare the Hubbard trees, see Figs. 4 and 31 for an example), hence satisfied for all c ∈ ∂M, so Φ_{pp′}^{q}(∂M_{c_{0}}) = ∂M_{c′_{0}}. The proof is completed through the following topological lemma:

Lemma 5.28 Let X_{0} ⊂ X ⊂ ℂ and X′_{0} ⊂ X′ ⊂ ℂ. Suppose X_{0} and X′_{0} are compact, connected and full in ℂ and satisfy ∂X_{0} ⊂ ∂X and ∂X′_{0} ⊂ ∂X′. If ϕ : X ⟶ X′ is a homeomorphism satisfying ϕ(∂X_{0}) = ∂X′_{0} then ϕ(X_{0}) = X′_{0}.

q.e.d.

6 Involutions: Proof of Theorem B

If we look at the pictures of the connectedness loci L_{q} (see Figs. 10 to 15), we observe a symmetry with respect to complex conjugation. Indeed, the function

\begin{aligned}C : & L_{q} & ⟶ & L_{q} \\ & λ & ⟼ & \overline{λ} \end{aligned}

is a homeomorphism, and C^{2} = Id. The map C maps the 0-limb to itself and the polynomials P_{q,λ} and P_{q,\overline{λ}} are conjugate through the anti-holomorphic map C : z ↦ \overline{z}.

It is then clear that the function

I_{p/q} : M_{p/q} ⟶^{ϕ_{p/q}} L_{q,0} ⟶^{C} L_{q,0} ⟶^{ϕ_{p/q}^{−1}} M_{p/q}

is a non-trivial homeomorphism from the p/q-limb of M to itself, such that I_{p/q}^{2} = Id.

Since ϕ_{p/q} is holomorphic in the interior of M_{p/q} it follows that I_{p/q} is anti-holomorphic in the interior of M_{p/q}.

From the construction, we see that I_{p/q} is like a reflection with respect to an arc of symmetry. This arc is the preimage of the real axis (up to the “last” Misiurewicz point in L_{q,0}) under ϕ_{p/q}, and hence it is always a topological arc (see Figs. 5 and 6).

The involution I_{p/q} can also be obtained by using the symmetry of the Mandelbrot set and the homeomorphisms between limbs given by Theorem A. The map C maps the limb M_{p/q} onto the limb M_{(q−p)/q} and the polynomials Q_{c} and Q\overline{c} are conjugate through the anti-holomorphic map C : z ↦ \overline{z}. We have the following lemma:

Lemma 6.1 The homeomorphisms ϕ_{p/q} : M_{p/q} ⟶ L_{q,0} are compatible with complex conjugation, i.e. the diagram

M_{p/q}  ⟶^{ϕ_{p/q}} L_{q,0}
C↓  ↓^{C}
M_{(q−p)/q} ⟶^{ϕ_{(q−p)/q}} L_{q,0}

commutes.

Proof : For this proof we use notation from section 5.2. Fix the potential η > 0 and the slope s. The first return map satisfies f (1) ; \overline{c}(z) = \overline{f_{c}^{(1)}(\overline{z})}. We can then modify the maps in

a similar fashion, i.e. set f (2) ; \overline{c}(z) = \overline{f_{c}^{(2)}(\overline{z})}. The map φ\overline{c}(z) = \overline{φ_{c}(\overline{z})} is an integrating map and it defines the polynomial-like map f (3) ; \overline{c} to be

f (3) ; \overline{c}(z) = \overline{f_{c}^{(3)}(\overline{z})}

Finally, the map χ\overline{c}(z) = \overline{χ_{c}(\overline{z})} is a hybrid equivalence between f (3) ; \overline{c} and P_{q,\overline{λ}}.

q.e.d.

From the lemma above it follows that

I_{p/q} = ϕ_{p/q}^{−1} ∘ C ∘ ϕ_{p/q} = ϕ_{p/q}^{−1} ∘ ϕ_{(q−p)/q} ∘ C = Φ_{(q−p)p}^{q} ∘ C
= C ∘ ϕ_{(q−p)/p}^{−1} ∘ ϕ_{p/q} = C ∘ Φ_{p(q−p)}^{q}.

Note that without reference to the polynomials P_{q,λ} it is not clear that the fixed points under I_{p/q} form a topological arc.

Remark 6.2 The map I_{p/q} maps hyperbolic components to hyperbolic components and the corresponding multiplier maps are complex conjugates of each other. •

Very much in particular, and as an example, the map I_{1/3} gives a homeomorphism between the two main antennas in the 1/3-limb, sending c = i to the landing point of R_{M}(1/4), and the center of the hyperbolic component of period 4 to the center of the hyperbolic component of period 5.

The dynamical statement of Theorem B follows similarly. Set I = I_{p/q}. On one hand, for c ∈ M_{p/q} and λ = ϕ_{p/q}(c) we can define

\begin{aligned}Î_{c}^{T} : & N(K_{c}^{T}) & ⟶^{H_{c}} & N(K_{λ}) & ⟶^{C} & N(K\overline{λ}) & ⟶^{H_{I(c)}^{−1}} & N(K_{I(c)}^{T}) \\ & z & ⟼ & H_{c}(z) & ⟼ & \overline{H_{c}(z)} & ⟼ & H_{I(c)}^{−1}(\overline{H_{c}(z)}) \end{aligned}

where H_{c} and H_{I(c)} are as in Theorem G. The map Î_{c}^{T} is an orientation reversing homeomorphism between N(K_{c}^{T}) and N(K_{I(c)}^{T}) (which in fact conjugates the two first return

maps f_{c}^{(1)} and f_{I(c)}^{(1)} on K_{c}^{T} and K_{I(c)}^{T} respectively). The map Î_{c}^{T} can be extended to a homeomorphism Î_{c} (similarly to the extension of Φ̂_{c}^{T} to Φ̂_{c}) between neighborhoods N(K_{c}) and N(K_{I(c)}) respectively, as stated in Theorem B.

On the other hand, it follows from the lemma above that the maps in dynamical plane are compatible with complex conjugation. That is, for any c ∈ M_{p/q} the following diagram commutes:

N(K_{c}) \underrightarrow{_{Φ_{c}}} N(K_{Φ_{p(q−p)}^{q}})
C↓  ↓^{C}
N(K\overline{c}) \underrightarrow{_{Φ\overline{c}}} N(K_{I_{p/q}(c)})

so Î_{c} : N(K_{c}) ⟶ N(K_{I_{p/q}(c)}) can alternatively be defined as

Î_{c} = Φ̂\overline{c} ∘ C = C ∘ Φ̂_{c}.

7 Combinatorial Surgery:

Proof of Theorems H, C, D and E

7.1 Proof of Theorems H and C

The main goal in this section is to prove Theorem C. The strategy is the same as for proving Theorem A, namely via the higher degree polynomials P_{q,λ} with λ ∈ L_{q,0}.

We first introduce the necessary notation and formulate Theorem H, the analogue of Theorem C building the bridge to the higher degree polynomials.

In Sect. 7.1.1 we prove the part of Theorem H which deals with the dynamical planes and deduce the analogous part of Theorem C.

In Sect. 7.1.2 we prove the part of Theorem H which deals with the parameter spaces and deduce the analogous part of Theorem C.

Let c ∈ M_{p/q} and λ = ϕ_{p/q}(c) ∈ L_{q,0}. Recall from Theorem G that there is a quasiconformal homeomorphism

H_{c} : X_{c} ⟶ W_{λ}

conjugating f_{c}^{(2)} and P_{λ}. In particular, it is uniquely defined on K_{c}^{T} and conjugates the first return map f_{c}^{(1)} and P_{λ} on the filled Julia sets.

In this section we extend this conjugacy to the exterior of the filled Julia sets in a “combinatorial way” which is explained below.

Let m_{d} : 𝕋 ⟶ 𝕋 denote the map induced on the circle 𝕋 = ℝ/ℤ by M_{d} : ℝ ⟶ ℝ, where M_{d}(x) = dx is multiplication by d.

Set 𝕀_{p/q} := [θ̃^{q−1}, θ̃^{0}] as a subset of 𝕋 (and hence with the metric of 𝕋). Define the truncated circle to be

𝕋^{T} = 𝕋_{p/q}^{T} := 𝕀_{p/q}/ ∼

where ∼ denotes the equivalence relation identifying the endpoints of 𝕀_{p/q}. Denote as before the identified argument by θ̃.

Let m_{2}^{(1)} = m_{p/q,2}^{(1)} : 𝕋_{p/q}^{T} ⟶ 𝕋_{p/q}^{T} denote the first return map of the doubling map m_{2}, i.e.

m_{2}^{(1)}(θ) = \begin{cases}2^{q}θ & if θ^{q−1} ≤ θ ≤ θ^{0} \\ 2^{q−1}θ & if θ^{p−1} ≤ θ ≤ θ^{p} \\ 2^{q−2}θ & if θ^{2p−1} ≤ θ ≤ θ^{2p} \\ … \\ 2θ & if θ^{q−p−1} ≤ θ ≤ θ^{q−p} \end{cases}

The first return map m_{2}^{(1)} is an orientation preserving, expanding covering map of degree q + 1, fixing θ̃. Moreover, consider

m_{q+1} : 𝕋 ⟶ 𝕋.

The map induced from multiplication by q + 1 is also an orientation preserving, expanding covering map of degree q + 1, fixing 0.

The lemma below is the tool we need in order to define the combinatorial extension of H_{c}.

Lemma 7.1 Let 𝕋_{j}, j = 1, 2 be a metric space homeomorphic to 𝕋 and with basepoint t_{j} ∈ 𝕋_{j}. Suppose E_{j} : 𝕋_{j} ⟶ 𝕋_{j}, j = 1, 2 is an orientation preserving, expanding covering map of degree d > 1, fixing t_{j}. Then, there exists a unique orientation preserving homeomorphism T : 𝕋_{1} ⟶ 𝕋_{2}, conjugating E_{1} and E_{2} and mapping t_{1} to t_{2}.

Hence, it follows that there exists a unique orientation preserving homeomorphism

Θ̂ = Θ̂_{p/q} := 𝕋_{p/q}^{T} ⟶ 𝕋  (11)

conjugating m_{2}^{(1)} and m_{q+1} and mapping θ̃ to 0. Note that Θ̂ is defined independently of c ∈ M_{p/q} and λ ∈ L_{q,0}. Proof of Lemma 7.1: We give a sketch of the proof. Choose any orientation preserving homeomorphism T_{0} : 𝕋_{1} ⟶ 𝕋_{2} which maps t_{1} to t_{2}. Define by induction T_{n} : 𝕋_{1} ⟶ 𝕋_{2} to be the homeomorphism which makes the diagram

𝕋_{1} ⟶^{E_{1}} 𝕋_{1}
T_{n}↓  ↓^{T_{n−1}}
𝕋_{2} ⟶^{E_{2}} 𝕋_{2}

commutative and which maps t_{1} to t_{2}. One can prove, using that E_{i} is expanding, that T_{n} converges to a homeomorphism T with properties as claimed.

q.e.d.

For a given p/q recall that θ_{p/q}^{−} = θ̃^{p−1} and θ_{p/q}^{+} = θ̃^{p} are the arguments of the external rays of M landing at the root point of the limb M_{p/q}. Moreover, recall that we have defined 0 and 1 to be the arguments relative to the 0-wake W L_{q,0} of the external rays of L_{q} landing at the root point λ = 1 of the 0-limb L_{q,0}. Note that the interval [θ_{p/q}^{−}, θ_{p/q}^{+}] = [θ̃^{p−1}, θ̃^{p}] is mapped homeomorphically onto the interval 𝕀_{p/q} = [θ̃^{q−1}, θ̃^{0}] by the (q − 1)-st iterate of the doubling map

m_{2}^{q−1} : [θ_{p/q}^{−}, θ_{p/q}^{+}] ⟶ 𝕀_{p/q}.

Define the homeomorphism

Θ_{p/q} : [θ_{p/q}^{−}, θ_{p/q}^{+}] ⟶ [0, 1]

as

Θ_{p/q} = Θ̂_{p/q} ∘ m_{2}^{q−1}  (12)

on the open interval (θ_{p/q}^{−}, θ_{p/q}^{+}).

Theorem H Given p/q, let Θ̂_{p/q} and Θ_{p/q} be as defined in eqs. (11) and (12) respectively.

1. Suppose θ = \frac{r}{s} ∈ [θ_{p/q}^{−}, θ_{p/q}^{+}] with gcd(r, s) = 1 and s even. Then the ray R_{M}(θ) lands at a point c ∈ M_{p/q} if and only if R_{L_{q,0}}(Θ_{p/q}(θ)) lands at ϕ_{p/q}(c) ∈ L_{q,0}.

2. Suppose c ∈ M_{p/q} and λ = ϕ_{p/q}(c) ∈ L_{q,0}. Recall that a hybrid equivalence H_{c} between f_{c}^{(3)} and P_{λ} is uniquely determined on K_{c}^{T}. For any θ ∈ 𝕋_{p/q}^{T} \{θ̃} the ray R_{c}(θ) lands at a point z ∈ K_{c}^{T} if and only if the ray R_{λ}(Θ̂_{p/q}(θ)) lands at H_{c}(z) ∈ K_{λ}.

7.1.1 Dynamical Plane

Let c ∈ M_{p/q} and λ = ϕ_{p/q}(c) ∈ L_{q,0}. Moreover let H_{c} : X_{c} ⟶ W_{λ} be given as in Theorem G.

Proposition 7.2 The map H_{c} induces an orientation preserving homeomorphism

Θ_{c} : 𝕋_{p/q}^{T} ⟶ 𝕋

satisfying:

1. Θ_{c}(θ̃) = 0.

2. For any θ ∈ 𝕋_{p/q}^{T} \ {θ̃}, the external ray R_{c}(θ) lands at a point z ∈ K_{c}^{T} if and only if the external ray R_{λ}(Θ_{c}(θ)) lands at H_{c}(z) ∈ K_{λ}.

3. Θ_{c} conjugates m_{2}^{(1)} and m_{q+1}, hence Θ_{c} = Θ̂_{p/q}.

From part 3 follows this corollary:

Corollary 7.3 Θ_{c} maps periodic (resp. preperiodic) arguments under m_{2}^{(1)} to periodic (resp. preperiodic) arguments under m_{q+1}.

Proof of 7.2: This would follow immediately from a stronger result called The Prime Ends Theorem ([Pom]). However, we give here a proof for this weaker version. In order to define the map Θ_{c} we need some preliminaries.

Given K a compact, connected and full subset of ℂ, a point z ∈ K is called accessible if there exists a curve γ : [0, ε] ⟶ ℂ with γ(0) = z and γ((0, ε]) ⊂ ℂ \ K. The homotopy class of such curves is call an access to z. If K is locally connected, then all points in K are accessible.

We shall need the following result:

Suppose z is accessible. Then, to each access [γ] there is associated a unique external ray R(θ) that lands at z and such that an arc of the ray belongs to the homotopy class [γ] (see [Pe, Mi]).

We define Θ_{c} first in the set of arguments that correspond to accesses. Let

A(K_{c}^{T}) = {θ ∈ 𝕋_{p/q}^{T} | θ corresponds to an access of K_{c}^{T} },
A(K_{λ}) = {θ ∈ 𝕋 | θ corresponds to an access of K_{λ}}

Note that if K_{c}^{T} (and hence K_{λ}) are locally connected, then A(K_{c}^{T}) = 𝕋_{p/q}^{T} and A(K_{λ}) = 𝕋.

Definition of Θ_{c} on A(K_{c}^{T}) Given θ ∈ A(K_{c}^{T}), let γ(θ) be the curve which is the connected component of \overline{R_{c}(θ)}∩N(K_{c}^{T}) containing z = R_{c}^{∗}(θ) (the landing point of R_{c}(θ)). Then, H_{c}(γ(θ)) is a curve landing at H_{c}(z). Define Θ_{c}(θ) as the external argument of the external ray associated with [H_{c}(γ(θ))], R_{λ}(Θ_{c}(θ)). Rays landing at H_{c}(z) have arguments in A(K_{λ}), so Θ_{c}(θ) ∈ A(K_{λ}).

Note that Θ_{c}(θ̃) = 0.

Lemma 7.4 The map Θ_{c} is an orientation preserving bijection between A(K_{c}^{T}) and A(K_{λ}).

Proof : To show injectivity, take two elements in A(K_{c}^{T}), θ ≠ θ′ and first suppose that R_{c}(θ) and R_{c}(θ′) land at different points. Then, since H_{c} is a homeomorphism, R_{λ}(Θ_{c}(θ)) and R_{λ}(Θ_{c}(θ′)) must land at different points. Hence Θ_{c}(θ) ≠ Θ_{c}(θ′). Now suppose that R_{c}(θ) and R_{c}(θ′) land at the same point. Then γ(θ) and γ(θ′) are not homotopic which implies that H_{c}(γ(θ)) and H_{c}(γ(θ′)) are not homotopic. Hence Θ_{c}(θ) ≠ Θ_{c}(θ′).

To show surjectivity, let R_{λ}(θ′) be a ray landing at z′ ∈ K_{λ} so that θ′ ∈ A(K_{λ}), and let γ(θ′) be the curve as in the definition above. Then, H_{c}^{−1}(γ(θ′)) is a curve landing at H_{c}^{−1}(z′). Let θ ∈ A(K_{c}) be the external argument of the ray associated with [H_{c}^{−1}(γ(θ′))]. It is clear that Θ_{c}(θ) = θ′, since the homotopy classes are preserved by the homeomorphism.

The fact that Θ_{c} is orientation preserving follows from H_{c} being orientation preserving. q.e.d.

Corollary 7.5 For every accessible point z ∈ A(K_{c}^{T}), the number of rays landing at z equals the number of rays landing at H_{c}(z) ∈ K_{λ}.

The map Θ_{c} is now defined from a dense set in 𝕋_{p/q}^{T} to a dense set in 𝕋. Note that we can also consider it as a map from a dense set in [θ̃^{q}, θ̃^{0}] to a dense set in [0, 1] by defining

Θ_{c}(θ̃^{q}) = 0 and Θ_{c}(θ̃^{0}) = 1.

The next step is to extend it to the whole interval.

If K_{c}^{T} is locally connected, then Θ_{c} is already defined in the whole interval. Moreover, since it is bijective and order preserving between two compact sets of ℝ, it follows that Θ_{c} is a homeomorphism and we are done.

If K_{c}^{T} is not locally connected we will use the following topological lemma to extend Θ_{c}.

Lemma 7.6 Let S be a dense set in an interval [a, b], and τ : S ⟶ [c, d] an order preserving injective function. Assume also that τ(S) is dense in [c, d]. Then, τ has a unique extension to a homeomorphism τ : [a, b] ⟶ [c, d].

The map Θ_{c} : A(K_{c}^{T}) ⟶ A(K_{λ}) satisfies all the hypothesis of Lemma 7.6. Abusing notation, we obtain the unique extension

Θ_{c} : [θ̃^{q}, θ̃^{0}] ⟶ [0, 1] or Θ_{c} : 𝕋 ⟶ 𝕋,

which is an orientation preserving homeomorphism. By definition, this map satisfies the second statement of Prop. 7.2.

To prove the last statement first suppose that θ belongs to A(K_{c}^{T}), z = R_{c}^{∗}(θ) ∈ K_{c}^{T}. Then Θ_{c}(m_{2}^{(1)}(θ)) is the argument of a ray landing at H_{c}(f_{c}^{(1)}(z)) ∈ K_{λ}. On the other hand m_{q+1}(Θ_{c}(θ)) is the argument of a ray landing at P_{λ}(H_{c}(z)). Since H_{c} conjugates f^{(1)} to P_{λ}, these two points are the same. It follows that Θ_{c} conjugates m_{2}^{(1)} to m_{q+1} at least up to arguments of rays landing at the same point. Therefore, we must show that equivalence classes of rays landing at the same point cannot be permuted, but this follows from the fact that Θ_{c}, m_{2}^{(1)} and m_{q+1} are order preserving for arguments that belong to the same V_{c}^{i} (in particular, for rays in the same equivalence class) and from Θ_{c}(θ̃) = 0.

Therefore, the map Θ_{c} conjugates m_{2}^{(1)} and m_{q+1} on a dense set and hence everywhere. By Lemma 7.1 such a homeomorphism is unique and therefore Θ_{c} = Θ̂_{p/q}. This ends the proof of Theorem 7.2

q.e.d.

We proceed now to combine the maps above to construct

(Θ̂_{pp′}^{q})^{T} : 𝕋_{p/q}^{T} \underrightarrow{_{Θ_{p/q}}} 𝕋 \underrightarrow{_{Θ_{p′/q}^{−1}}} 𝕋_{p′/q}^{T}.

The next proposition follows from Prop. 7.2:

Proposition 7.7 Let c ∈ M_{p/q} and c′ = Φ_{pp′}^{q}(c) ∈ M_{p′/q}. Then,

1. A ray R_{c}(θ) lands at z ∈ K_{c}^{T} if and only if the ray R_{c′}((Θ̂_{pp′}^{q})^{T} (θ)) lands at Φ̂_{c}^{T} (z).

2. The map (Θ̂_{pp′}^{q})^{T} conjugates the first return maps on the arguments in 𝕋_{p/q}^{T} and 𝕋_{p′/q}^{T} respectively, that is the following diagram commutes:

𝕋_{p/q}^{T} ⟶^{m_{p/q,2}^{(1)}} 𝕋_{p/q}^{T}
(_{Θ_{pp′}^{q}})^{T} ↓  ↓^{(_{Θ_{pp′}^{q}})^{T}}
𝕋_{p′/q}^{T} ⟶^{m_{p′/q,2}^{(1)}} 𝕋_{p′/q}^{T}

Finally, we extend the map (Θ̂_{pp′}^{q})^{T} to the rest of the arguments of the filled Julia set to obtain the map claimed in part two of Theorem C. Note that this is a somewhat “artificial” extension, since we will not use the dynamics of the polynomial. However, it is a nice complement to the map Φ̂_{c} on the filled Julia sets.

As before let c ∈ M_{p/q} and c′ = Φ_{pp′}^{q}(c) ∈ M_{p′/q}.

Definition We define

Θ̂_{pp′}^{q} : 𝕋 ⟶ 𝕋

by

Θ̂_{pp′}^{q}(θ) = \begin{cases}(Θ̂_{pp′}^{q})^{T} (θ) & if θ ∈ 𝕀^{∘}_{p/q} \\ (Θ̂_{pp′}^{q})^{T} (θ + \frac{1}{2}) + \frac{1}{2} & otherwise \end{cases}

Note that this map is well defined since the arguments of external rays in V^{∘} _{c}^{0} satisfy

(Θ̂_{pp′}^{q})^{T} (θ) = (Θ̂_{pp′}^{q})^{T} (θ + \frac{1}{2}) + \frac{1}{2}

The map Θ̂_{pp′}^{q} is an orientation preserving homeomorphism satisfying part two of Theorem C, that is:

Proposition 7.8 A ray R_{c}(θ) lands at z ∈ K_{c} if and only if the ray R_{c′}(Θ̂_{pp′}^{q}(θ)) lands at Φ̂_{c}(z) ∈ K_{c′}.

Remark 7.9 We could have deduced the map on arguments directly from the homeomorphism ϕ̂_{c} : N(K_{c}) ⟶ N(K_{c′}). Instead, we have once again applied the bridge to the higher degree polynomials in order to show that the map Θ̂_{pp′}^{q} only depends on p, p′ and q. •

Proposition 7.10 Suppose K_{c} (and therefore K_{c′}) is locally connected. Then, there exists a homeomorphism ˆ ϕ̂_{c} which coincides with ϕ̂_{c} on K_{c} and which maps the external ray R_{c}(θ) to the external ray R_{c′}(Θ̂_{pp′}^{q}(θ)) equipotentially.

Remark 7.11 If K_{c} is not locally connected, we can still define the homeomorphism from ℂ \ K_{c} to ℂ \ K_{c′} as above, but this map may not match continuously with ϕ̂_{c} restricted to K_{c}. •

7.1.2 Parameter Plane

We are now ready to prove the combinatorial part of the parameter space theorems.

We start by proving the first part of Theorem H. Then we obtain the first part of Theorem C for rational arguments with even denominators. Finally, we extend it to all rational arguments. Proof of the first part of Theorem H: Let Θ_{p/q} = Θ̂_{p/q} ∘ m_{2}^{q−1} : (θ_{p/q}^{−}, θ_{p/q}^{+}) ⟶ (0, 1) as defined earlier.

Suppose c is a Misiurewicz point in M_{p/q}. Recall that

R_{c}(θ) lands at c ∈ K_{c} ⟺ R_{M}(θ) lands at c ∈ M_{p/q}.

Moreover

R_{c}(θ) lands at c ∈ K_{c} ⟺ R_{c}(2^{q−1}θ) lands at Q_{c}^{q−1}(c) = Q_{c}^{q}(0) ∈ K_{c}.

Note that Q_{c}^{q}(0) is the critical value of f_{c}^{(3)}, hence mapped by H_{c} to the critical value v_{λ} = λ(−1)^{q}(\frac{q}{q+1})^{q+1} of P_{λ} where λ = ϕ_{p/q}(c).

It follows from the dynamical part of Theorem H that

R_{c}(2^{q−1}θ) lands at Q_{c}^{q}(0) ∈ K_{c} ⟺ R_{λ}(Θ̂_{p/q}((2^{q−1}θ)) lands at v_{λ} ∈ K_{λ}.

Finally, apply Prop. 3.9 to obtain that

R_{λ}(Θ̂_{p/q}((2^{q−1}θ)) lands at v_{λ} ∈ K_{λ} ⟺ R_{L_{q,0}}(Θ̂_{p/q}((2^{q−1}θ)) lands at λ ∈ L_{q,0}.

q.e.d.

Define

Θ_{pp′}^{q} : [θ_{p/q}^{−}, θ_{p/q}^{+}] ⟶^{Θ_{p/q}} [0, 1] ⟶^{Θ_{p′/q}^{−1}} [θ_{p′/q}^{−}, θ_{p′/q}^{+}].

The first part of Theorem C concerning rational arguments with even denominators, that is arguments of rays landing at Misiurewicz points, follows immediately from Theorem H.

The missing part is to prove part one of Theorem C for rational arguments with odd denominators, that is arguments of rays landing at roots of hyperbolic components.

Remark 7.12 From Cor. 7.3 it follows that Θ_{pp′}^{q} maps rational angles with even (resp. odd) denominator to rational angles with even (resp. odd) denominators. •

Lemma 7.13 Let c_{0} ∈ M_{p/q} be the root of a hyperbolic component and θ_{0} ∈ 𝕋 the argument of one of the rays landing at c_{0}. Then, there exists a sequence of Misiurewicz points c_{n} → c_{0} and a sequence of arguments θ_{n} → θ_{0} such that R_{M}^{∗}(θ_{n}) = c_{n}.

For a proof, see [H, S].

Fix c_{0}, θ_{0}, c_{n}, and θ_{n} as in the lemma. Let c′_{0} = Φ_{pp′}^{q}(c_{0}), c′_{n} = Φ_{pp′}^{q}(c_{n}), θ′_{0} = Θ_{pp′}^{q}(θ_{0}) and θ′_{n} = Θ_{pp′}^{q}(θ_{n}). By the remark above, θ′_{0} is rational with odd denominator. By continuity of Θ_{pp′}^{q} and Φ_{pp′}^{q} we have that θ′_{n} → θ′_{0} and c′_{n} → c′_{0}. By remarks 5.7 and 5.5, c′_{n} are Misiurewicz points and c′_{0} is the root of a hyperbolic component. It follows immediately from above that R_{M}^{∗}(θ′_{n}) = c′_{n}.

To conclude the proof of Theorem C we must show

R_{M}^{∗}(θ′_{0}) = c′_{0}.  (13)

We recall the definition of the impression of an external ray of a compact, connected, full subset K ⊂ ℂ (see [S]). Definition For any ϵ > 0 and any θ ∈ 𝕋 let I_{K,ϵ}(θ) denote the closed subset of ∂K given by

I_{K,ϵ}(θ) = \overline{R_{K}([θ − ϵ, θ + ϵ])} \ R_{K}([θ − ϵ, θ + ϵ]).

The impression I_{K}(θ) is defined as

I_{K}(θ) = ⋂_{ϵ>0} I_{K,ϵ}(θ).

Lemma 7.14 c′_{0} ∈ I_{M}(θ′_{0}).

Proof : For each ϵ > 0, there exists n_{ϵ} ∈ ℕ such that θ′_{n} ∈ [θ′_{0} − ϵ, θ′_{0} + ϵ] for all n ≥ n_{ϵ}. It follows that c′_{n} ∈ I_{M,ϵ}(θ′_{0}) for all n ≥ n_{ϵ}, therefore that c′_{0} ∈ I_{M,ϵ}(θ′_{0}) for all ϵ > 0. Hence c′_{0} ∈ I_{M}(θ′_{0}).

q.e.d.

The conclusion will follow from the following lemma.

Lemma 7.15 Let c ∈ M be the root of a hyperbolic component Ω and θ a rational angle with odd denominator. If c ∈ I_{M}(θ) then I_{M}(θ) = {c}.

Proof : There are two cases, depending on Ω being a satellite component or a primitive component. The proof in the satellite case and “half” of the primitive case can be found in [S]. The missing part is the following, using a technique from [DH2], part II.

Suppose Ω (different from Ω_{0}) is primitive and θ^{±} are the rational arguments of the rays landing at c. Then, there exists a sequence of Misiurewicz points on the combinatorial vein in M between 0 and c, and sequences of arguments θ′_{n} and θ′′_{n}, where R_{M}^{∗}(θ′_{n}) = R_{M}^{∗}(θ′′_{n}) = c_{n} and θ′_{n} ↗ θ^{−} and θ′′_{n} ↘ θ^{+}. The existence is proved by modifying the Hubbard tree for the center of Ω to become the Hubbard tree of Misiurewicz points as described, transferring arguments from the dynamical plane to the parameter plane as usual. (See also [L, Poi, T]).

q.e.d.

With this lemma, (13) follows immediately.

Remark 7.16 The result above can be carried over by the same technique to the set of irrational arguments that correspond to rays that land at c-values on boundaries of hyperbolic components and also to c-values described by Yoccoz para-puzzles. •

7.2 Proof of Theorem D

In this section we combine the symmetries of the 0-wakes W L_{q,0} with respect to complex conjugation and Theorem C to obtain a non trivial homeomorphism from the set of arguments of rays in a p/q-wake W M_{p/q} to itself. This map is the combinatorial analogue of the involutions I_{p/q} in Theorem B.

Recall from section 6 that the map

\begin{aligned}C : & W L_{q,0} & ⟶ & W L_{q,0} \\ & λ & ⟼ & \overline{λ} \end{aligned}

is a homeomorphism mapping the ray R_{L}(θ) to the ray R_{L}(1 − θ) for θ ∈ (0, 1). We define the orientation reversing homeomorphism from [θ_{p/q}^{−}, θ_{p/q}^{+}] onto itself by

\begin{aligned}\overline{Θ}_{p/q} : & [θ_{p/q}^{−}, θ_{p/q}^{+}] & ⟶^{Θ_{p/q}} & [0, 1] & ⟶ & [0, 1] & ⟶^{Θ_{p/q}^{−1}} & [θ_{p/q}^{−}, θ_{p/q}^{+}] \\ & θ & ↦ & Θ_{p/q}(θ) & ↦ & 1 − Θ_{p/q}(θ) & ↦ & Θ_{p/q}^{−1}(1 − Θ_{p/q}(θ)) \end{aligned}

Note that this is an orientation reversing homeomorphism from the arguments of the rays in W M_{p/q} to themselves. It also satisfies \overline{Θ}_{p/q}^{2} = Id. In particular, the arguments that bound the wake are mapped to each other.

The fixed point of \overline{Θ}_{p/q} is θ^{s} := Θ_{p/q}^{−1}(1/2), hence rational.

This proves properties 1 and 2. It is clear that by the properties of Θ_{p/q}, property 3 is satisfied, i.e.

R_{M}^{∗}(θ) = c ⟺ R_{M}^{∗}(\overline{Θ}_{p/q}(θ)) = I_{p/q}(c).

For property 4 in dynamical plane we define

\begin{aligned}ˆ\overline{Θ}_{p/q}^{T} : & I_{p/q} & \underrightarrow{_{Θ_{p/q}}} & [0, 1] & ⟶ & [0, 1] & \underrightarrow{_{Θ_{p/q}^{−1}}} & I_{p/q} \\ & θ & ⟼ & Θ̂_{p/q}(θ) & ⟼ & 1 − Θ̂_{p/q}(θ) & ⟼ & Θ̂_{p/q}^{−1}(1 − Θ̂_{p/q}(θ)) \end{aligned}

Then, as we did in section 7.1.1 to define Θ_{pp′}^{q}, we use the symmetries of K_{c} with respect

to the origin to extend ˆ \overline{Θ}_{p/q}^{T} to an orientation reversing homeomorphism ˆ \overline{Θ}_{p/q} : 𝕋 ⟶ 𝕋 satisfying the required properties.

This concludes the proof of Theorem D.

7.3 Proof of Theorem E

It is clear that we can use the map Θ_{pp′}^{q} in Theorem C to define a homeomorphism from W M_{p/q} \ M_{p/q} to W M_{p′/q} \ M_{p′/q} by mapping a ray of argument θ to the ray of argument Θ_{pp′}^{q}(θ) equipotentially. By the properties of Θ_{pp′}^{q}, this homeomorphism would match with Φ_{pp′}^{q} at all the landing points of rays with rational arguments (together with the landing points of rays with irrational arguments covered by remark 7.16).

Now suppose the Mandelbrot set were locally connected. It would follow that all external rays of M land continuously and therefore, the radial extension would match continuously with Φ_{pp′}^{q} at all points in the boundary of M_{p/q}.

The extension of I_{p/q} to the p/q-wake would follow immediately by composition of the extensions of Φ_{p(q−p)}^{q} and complex conjugation.

This concludes the proof of Theorem E.

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