The Fine Structure of Herman Rings

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura's quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen's results on the fine local geometry of Siegel disks to the Herman ring setting.


Introduction
We consider the dynamical system induced by the iterates of a rational map f : C → C of degree d ≥ 2, where C denotes the Riemann sphere or compactified complex plane. We use the notation f n := f • n) · · · • f to denote the n th iterate of f . Under this dynamics, the Riemann sphere splits into two completely invariant sets: the Fatou set, formed by those points for which the sequence {f n } is normal in some neighborhood; and its complement, the Julia set. By definition the Fatou set is open and therefore the Julia set is a compact set of the sphere. Connected components of the Fatou set, also known as Fatou components, map onto one another and are eventually periodic [Sul85]. The Julia set is the common boundary between the different Fatou components and, consequently, the dynamics on this set is chaotic. For background on the dynamics of rational maps we refer for example to [CG93] and [Mil06].
An especially relevant particular case of rational maps are polynomials, which are exactly (up to Möbius conjugation) those rational maps for which infinity is a fixed point and has no preimages other than itself. In particular this implies that infinity is a superattracting fixed point, and the dynamics are locally conjugate to z → z d around this point for some d ≥ 2, the degree of the polynomial; it also means that the basin of attraction of infinity, that is the set of points attracted to infinity under iteration, is connected and completely invariant. Therefore its boundary is compact in C and coincides with the Julia set of the polynomial.

Setup and statement of results
Arithmetics play an important role in dynamics as well. It is important to distinguish between three nested classes of irrational numbers. For a 1 , a 2 , . . . ∈ N, we let [a 1 , a 2 , . . .] = 1 a 1 + 1 a 2 + . . . denote the continued fractional expansion with a 1 , a 2 , . . . as coefficients, and denote the convergents by p n q n = [a 1 , a 2 , . . . , a n ]. (1) See [Khi97] for details. An irrational number θ is a quadratic irrational if the sequence of coefficents a 1 , a 2 , . . . is eventually periodic. The quadratic irrationals are exactly the irrational roots of quadratic equations with integer coefficients. A more general set of irrational numbers are the irrationals of bounded type. They are numbers whose coefficients satisfy sup a n < ∞.
An even more general class of irrationals is the class of Brjuno numbers which we denote by B. A Brjuno number is charaterized by the denominators of its convergents; a number is Brjuno if and only if log q n+1 /q n < ∞. These classes of irrational numbers are relevant to dynamics. If f is a holomorphic map in a neighborhood of the origin, such that f (0) = 0 and f (0) = e 2iπα with α ∈ B, then there is a neighborhood of 0 on which f is conjugate to an irrational rotation of rotation α [Sie42, Brj65,Rüs67]. If the map is globally defined, this is part of a Siegel disk. Conversely, if a quadratic polynomial has an invariant Siegel disk, its center is a fixed point with multiplier e 2iπα with α ∈ B [Yoc95].
Here and in the rest of the article, we fix an irrational number θ of bounded type. We let λ = e 2iπθ , and fix the quadratic polynomial This polynomial has a unique critical point ω := − λ 2 . The origin is a fixed point of multiplier P (0) = λ. We know that P posseses a Siegel disk S centered at z = 0, because the numbers of bounded type form a subset of the Brjuno numbers.
On the rational end, we work with the simplest family that can exhibit Herman rings, namely for a, b ∈ C. Every f a,b has superattracting fixed points at the origin and at infinity. Additionally, there are two other criticial points which we denote by ω 1 and ω 2 . The family f a,b provides examples of Herman rings. It is well known [Brj65,Shi87,Yoc95] that for any irrational α, there exists a, b such that f a,b has a Herman ring with rotation number α, if and only if α is Brjuno. In [BFGH05], Buff, Fagella, Geyer and Henriksen show that for a Brjuno number α, there exists a pointed disk holomorphically embedded in the a, b parameter space of f a,b , such that every mapping in the disk possesses an invariant Herman ring with rotation number α.
From now on, we let a, b be parameters chosen such that f a,b has an invariant Herman ring with rotation number θ. Since we shall not vary a, b, we drop the indexes and simply denote the rational map by f . Hence has a Herman ring of rotation number θ, which we denote by H. The Herman ring has two boundary components ∂ j H, j = 1, 2, which are both quasicircles, each containing a critical point ω j (this follows from results of Herman, Ghys, Douady,Światek and Shishikura, see e.g. [BF14, Sections 7.2 and 7.3]). We number the components such that ∂ 1 H is contained in the bounded component of the complement of H, and we number the critical points, such that ω j ∈ ∂ j H, j = 1, 2.
As mentioned, f has a fixed critical point at infinty. Dynamically this means that infinity is a superattracting fixed point, the immediate attracting bassin of which we denote by A f (∞). The boundary of A f (∞) is a proper subset of the Julia set J(f ). This is in contrast to what happens for P , the quadratic polynomial in (2), where the boundary of the basin of infinity, A P (∞), coincides with J(P ).
In [Shi86], Shishikura introduced a surgery that could turn a map with a cycle of Herman rings into a map with a cycle and Siegel disk and viceversa (see Section 3). We will use a special case of his construction to show the first of our two main theorems.
Theorem A. Let θ be of bounded type and let P and f be as in (2) and (3) respectively. In the setup above, there exists a P -invariant simply connected domain D S, and a quasiconformal mapping Φ : C → C, such that  The notion of C 1+α -conformal is due to McMullen and is defined as follows.
Note that this is stronger that saying that φ is C−differentiable at the point z 0 . We say φ is C 1+α -anticonformal at z 0 , ifφ is C 1+α -conformal at z 0 .
Theorem A is illustrated in Figure 1. The fact that the quasiconformal conjugacy provided by Theorem A is C 1+α at ω allows us to see that the boundary components of H are locally similar to the boundary of S. To make a precise statement, we introduce a notion of similarity that is stronger than Tan Lei's notion of asymptotic similarity introduced in [Tan90]. Let B(c, r) denote the open ball of center c ∈ C and radius r > 0.
Definition 1.2 (Tight similarity). We say that two compact sets A, B are tightly similar at z 0 if there exists δ, β > 0 and L > 0 such that Figure 1: We illustrate Theorem A for θ = √ 5−1 2 , the golden mean. The dynamics of P is illustrated in the figure to the left, which is symmetric with respect to ω. We have colored the completely invariant set D in red, and J(P ) in black. The dynamics of f is illustrated in the figure to the right, which has been rotated, scaled and translated to illustrate the similarity with P . The Herman ring and its preimages are colored yellow and J(f ) black. By Theorem A, there exists a quasiconformal homeomorphism Φ which is conformal outside the red set. Outside D, Φ conjugates P to f , and Φ is C 1+α -conformal at ω.
When A and B are tightly similar at z 0 , we write It is easy to check that the notion of tight similarity at z 0 is an equivalence relation on the compact subsets of C. We are now ready to state the second main theorem.
Theorem B. Let θ be of bounded type and let P and f be as in (2) and (3) respectively. In the setup above, the following are satisfied.
(a) There exists a scaling factor L ∈ C \ {0} such that Theorem B is illustrated in Figure 2. There we can see the simililary between J(P ), ∂A f (∞) and J(f ). Even though J(f ) and ∂A f (∞) are topologically very different, the components of J(f ) \ ∂A f (∞) have increasingly small diameters, and get increasingly close to A f (∞) as we zoom in at ω 2 .
Theorem B can be applied when searching for mappings in f a,b with Herman rings of bounded type rotation number. Indeed, if q n denotes the denominator of the convergents to θ defined in (1), from Theorem A we know for j = 1, 2. Therefore the points (a, b) where f a,b has a Herman ring with rotation number θ satisfy when n is large, which narrows the search to a one-complex-dimensional set of parameters.
Let us finally note that we can use the dynamics to extend the results to any point ω that under iteration goes to ω, and any point u that is eventually mapped to either ω 1 or ω 2 . Corollary 1.3. Suppose that ω and u satisfy P n (ω ) = ω and f m (u) ∈ {ω 1 , ω 2 }. Then, and u ∈ ∂H , then ∂H is tightly self-similar around u. . Using this result, one can see (not without some work) that Theorems A and B also extend to a more general setting than the model family considered above. More precisely, if g is a rational like map that straightens to a member of the model family f a,b having a Herman ring of rotation number θ (an irrational of bounded type) then our results apply to the small Julia set of g.

Preliminaries about P and its Siegel disk
Before proving our two main theorems, we review some facts about the Julia set J(P ) and the Siegel disk S. Recall that P (z) = e 2iπθ z + z 2 , with θ of bounded type, and ω = − e 2πiθ 2 , the critical point. In [Pet96], Petersen showed the following theorem.
Theorem 2.1. The Julia set J(P ) is a locally connected set of zero Lebesgue measure.
McMullen proved a string of geometric results in [McM98]. He showed that J(P ) has Hausdorff dimension strictly less than two. He also showed that ω is a density point of the filled Julia set of P . In fact, he showed an even stronger result, namely that ω is a measurable deep point in a subset of the filled in Julia set K(P ) := C \ A P (∞).
Definition 2.2 ((Measurable) deep point). Let z 0 ∈ C and E be a Borel set. For r > 0, let s(r) be the largest radius so that B(z, s(r)) ⊂ B(z 0 , r) \ E, for some z ∈ C. We say that z 0 is a deep point in E if there exists α > 0 such that s(r) ≤ r 1+α , for all r small enough.
We call z 0 a measurable deep point in E, if there exists constants M, β, δ > 0 such that It is obvious that measurable deep implies deep. Observe also that if E ⊂ E are Borel sets and z 0 is a measurable deep point in E then z 0 is a measurable deep point in E .
Then ω is a measurable deep point in S .
If φ : C → C is quasiconformal and the support of µ φ gets thin close to a point z 0 , we can expect φ to be regular at z 0 . There are several results in this direction (see e.g. [LV73, Chapter 6]), and we will find use for the following theorem of McMullen.
Remark 2.5. The last conclusion, i.e. the derivative being nonzero, is implicitely used in McMullen's text although not explicitely stated. For completeness, let us show how it follows from the condition of measurable depth. In [LV73, Lemma 6.1] it is proven that if the dilatation of φ, say D(z), satisfies that the integral |z|<r D(z) − 1 |z| 2 dσ is convergent for every r, then φ is complex differentiable at 0 with nonzero derivative. By breaking the disk of radius r into a series of annuli A n = { 1 2 n+1 < |z| < 1 2 n }, one can bound the integral on each annulus by 4(K−1)M 2 βn , where M and β are the constants given by the fact that 0 is a measurable deep point of Ω, and K is the bound on the dilatation of µ. The integral is thus bounded from above by a geometric series and hence finite.
McMullen also showed that when θ is a quadratic irrational, the Siegel disk is selfsimilar at the critical point ω. More precisely he showed the following.
Theorem 2.6 ([McM98, Theorem7.1]). Suppose θ is a quadratic irrational, and let s denote the periodicity of the coefficients of its continued fraction. Let P be as in (2) and S be its Siegel disk. Then, there exist α > 0 and a locally defined homeomorphism ψ, conjugating P qn to P qn+s on ∂S for n sufficiently large. More precisely, we have for some complex number κ with 0 < |κ| < 1.
It follows from the theorem, that when s is even, ψ is C 1+α -conformal at ω, and when s is odd, ψ is C 1+α -anticonformal.
Remark 2.7 (Tight self-similarity of S). We will see later (see Remark 4.4) that this implies that the Siegel disk is actually tightly self-similar. More precisely, if s is even, and, when s is odd, where κ is the scaling factor in Theorem 2.6.
With the preceeding theorem in hand, Buff and Henriksen [BH99] were able to prove that for some values of θ, such as the golden mean √ 5−1 2 , the Siegel disk S contains an Euclidean triangle with a vertex at the critical point.

Quasiconformal surgery and Proof of Theorem A
In this section we prove Theorem A. We shall see that the proofs of (a), (b) and (c). follow directly from a surgery construction due to Shishikura, whereas the last part can be derived by bounding the relative area of the support of the quasiconformal distorsion of Φ as we approach ω. The main idea of the surgery is simply to replace the dynamics in the hole of the Herman ring with an irrational rotation. In this way we obtain a quasiregular map F , which is quasiconformally conjugate to P . Letting Φ denote the conjugacy from P to F , we then check that it has the stated properties. Details are as follows (c.f. [Shi87] and [BF14, Section 7.2]).
We keep the notation from the setup in Section 1.1. Let φ 0 : H → {z : r < |z| < 1} denote the linearizing map, conjugating f to R θ : z → e 2iπθ z. Define three topological disks U 1 U 2 U 3 , such that ∂U 1 ⊂ H and ∂U 2 ⊂ H are f −invariant curves and U 3 is the polynomially convex hull of H which in this case is the complement of the unbounded component of the complement of H. The image under φ 0 of ∂U 1 is a circle, i.e., the boundary of a disk V 1 . Similarly, we define V 2 to be the disk whose boundary is φ 0 (∂U 2 ), and we let V 3 = B(0, 1). See Figure 3. Figure 3: Quasiconformal surgery to produce the polynomial P with a Siegel disk starting from the rational map f with the Herman ring H.
We modify and extend φ 0 as to define it on all of U 3 . Define φ 1 : U 3 → V 3 by letting it be equal to φ 0 on U 3 \ U 2 , by requiring that it maps (U 1 , 0) conformally to (V 1 , 0) and interpolating quasiconformally on the annulus U 2 \ U 1 , so we get a quasiconformal mapping φ 1 : U 3 → V 3 , which we shall use to paste the rigid rotation into the Herman ring.
We have set up the machinery to plug the hole of the Herman ring. Let This is a model of a quadratic polynomial, since the pole of f no longer exists and the global degree is now two. It is, however, only quasiregular. To remedy this, we will define an F −invariant Beltrami coefficient µ with bounded dilatation with the intention of applying the Measurable Riemann Mapping Theorem. This Beltrami coefficient is defined by pieces. We start by defining it in U 3 , by pulling back the standard Beltrami coefficient µ 0 = 0 under φ 1 , that is µ = φ * 1 (0) on U 3 or equivalently, µ(z) =∂φ 1 /∂φ 1 . Observe that µ is invariant by F | U3 by construction, and it has bounded dilatation, precisely that of φ 1 .
We can extend µ recursively to the backward orbit of U 3 , by letting µ := (F n ) * (µ) on F −n (U 3 ), for every n ≥ 1. Finally we can extend it to all of C by letting µ vanish outside the backwards orbit of U 3 .
Property (a). First notice that φ 1 •Φ is conformal on Ψ(U 3 ) and conjugates P to R θ on this domain. So Ψ(U 3 ) ⊂ S. By maximality of H, Ψ(U 3 ) = S, because if Ψ(U 3 ) were only a subdisk of S then H would not the maximal domain of linearization. By construction, Φ conjugates P to F everywhere, but since F = f except on U 2 , Φ conjugates P to f everywhere except on D. Finally, since U 2 U 3 we know D S.
Property (c). By construction,∂Ψ = 0 on the complement of U = ∪ ∞ n=0 f −n (U 2 ), and thus locally conformal on the complement of the closure of this set. Hence the inverse map Φ is locally conformal on the complement of the closure of D = Ψ( U ). The boundary of D consists of a countable number of closed real analytic curves as well as the boundary of J(P ). By Petersen's theorem, Theorem 2.1, the boundary of D has measure zero. Thus we can conclude∂Φ = 0 almost everywhere on the complement of D.
Property (d). In view of Theorem 2.4, it is enough to prove that ω is a measurable deep point in the complement of D. To show this, McMullen has done the heavy lifting by proving Theorem 2.3: ω is a measurable deep point in S . Since D is a definite distance away from ω and again using the J(P ) has measure zero, we deduce that ω is a measurable deep point of S \ (D ∪ J(f )). Hence it is enough to show that S \ (D ∪ J(f )) ⊂ (C \ D).
The invariant set D has two preimages; D itself and another one D . Choosing small enough, we can assume S does not meet D . Any point z ∈ S \ (D ∪ J(f )) lies in the Fatou set, and is thus eventually mapped into S. Clearly, z is not an element of D ∪ D . However, since S is forward invariant, none of the iterates of P can map z into D . So z is not an element of D. This concludes the proof of Theorem A.
In the course of the proof, we showed that ω is a measurable deep point C \ D. We shall use this later, so we formally state it. 4 Preliminaries about C 1+α -conformal mappings and tightly similar sets To prove Theorem B, we need to establish some elementary properties of C 1+α -conformal homeomorphisms and tightly similar sets. We see in this section that the two notions complement each other well.
First we prove that C 1+α regularity extend to inverses when the map in question is quasiconformal.
Proof. Let β, δ, and K be such that 1. and 2. in definition 1.2 holds for A and C.
To prove A ∼ z0 B, it is enough to prove that when b ∈ B ∩ B(z 0 , δ), we can find a ∈ A such that |a − b| ≤ K|b − z 0 | 1+β . But since b ∈ C, and A ∼ z0 C, we can find such a.
To prove B ∼ z0 C, it is enough to prove that when C ∈ C ∩ B(z 0 , δ), we can find b ∈ B such that |c − b| ≤ K|c − z 0 | 1+β . Since b ∈ C, and A ∼ z0 C, we can find such b ∈ A ⊂ B, satisfying the inquality.
We end this section by seeing that tight similarity at a point is preserved by homeomorphisms which are C 1+α -conformal at the point. Proposition 4.3. Let A be a compact set and φ a homeomorphism which is C 1+αconformal at z 0 .

Proof.
To see the first statement, notice that for any a ∈ A ∩ B(z 0 , δ), we have φ(a) = a + R(a). Hence for some M > 0. This is the first of the requirements of Definition 1.2. The second requirement follows by applying the same argument to φ −1 , which is C 1+α -conformal by Proposition 4.1.
We can deduce the second statement from the first. Since scaling A and B around z 0 does not change whether they are tightly similar or not, we can suppose φ (z 0 ) = 1. Then and the two image sets are tightly similar.
Remark 4.4. From this proposition, together with Theorem 2.6 , we can see that S is tightly self similar if θ is a quadratic irrational. Indeed, let s be the period of the continued fraction, and assume it is even. Let ψ be the homeomorpfism and κ the scaling factor in Theorem 2.6, which ensure the self-similarity of S. Then, the map gives the tight self-similarity between ∂S − ω and κ(∂S − ω) * , where (∂S − ω) * denotes the set obtained from first translating ∂S and then reflecting in the real axis.

Geometry invariance. Proof of Theorem B and Corollary 1.3
In this section we prove Theorem B. Most of the statements are fairly obvious by the results we have uncovered by now. The delicate part is to prove that ∂A f (∞) ∼ ω2 J(f ).
Recall that Φ denotes the quasiconformal homeomorphism given in Theorem A, conjugating P to f on C\D, which is conformal outside D = ∪ ∞ n=0 P −n (D). Our strategy will be to prove that J(P ) ∼ ω D. Since Φ is C 1+α -conformal at ω, Φ(J(P )) = ∂A f (∞), and we shall see that Φ(D) contains J(f ), this will be sufficient to prove the desired similarity by Propositions 4.2 and 4.3.
The idea in proving J(P ) ∼ ω D is that the area of components of D must quickly decrease as we approach ω. By bounded geometry, this means that the diameter of the components of D must also quickly decrease. Each connected component D of D is contained in a connected component F of the Fatou set F (P ). The modulus of the annulus F \ D is independent of which component D we are considering and so for any z ∈ D the distance d(z, ∂F ) is comparable to the diameter of D , i.e., quickly decreasing as we approach ω. This is the idea, it remains to fill in the details. Claim 2. There exists K 2 , K 3 ∈ (0, ∞) such that for any connected component D of D, where α ∈ D is eventually mapped to the Siegel point.
The claim is trivial, since π times in-radius squared gives a lower bound on the area, π times the out-radius squared gives an upper bound, and in-radius and out-radius are comparable by Claim 1.
There are only finitely many components D of D withr(D , α ) > r 0 . If there were infinitely many, then by Claim 2, there area of D would be infinite, but D is contained in the filled-in Julia set of P , and it is well known that the latter set is contained in B(ω, 2). Hence we can pick δ > 0 so that Consider an arbitrary z ∈ D ∩ B(ω, δ ). Such z lies in a component D of D, and we have chosen δ so that r :=r(D , α ) ≤ r 0 . Notice D ⊂ B(ω, |z − ω| + 2r). As |z − ω| < δ ≤ δ/2, and 2r ≤ 2r 0 ≤ δ , it holds that D ⊂ B(ω, δ). So Using Claim 2, we get Since r ≤ r 0 , and by the choice of r 0 , the left hand side is bounded from below by r γ 2 , and we get that r ≤ K 5 |z − ω| 1+β/2 , for some K 5 :=∈ (0, ∞). By Claim 3, d(z, J(P )) ≤ K 3 K 5 |z − ω| 1+β/2 which proves Claim 4.
We have just established one of the two requirements of Definition 1.2. The set D is a completely invariant closed set contaning more than two points, so J(P ) ⊂ D. Hence, the other requirement is automatically satisfied.
Having proven Claim 5, we have navigated the rough part of the proof; the rest is smooth sailing. By Theorem A, Φ(J(P )) = ∂A f (∞). Again using Theorem A we obtain that L := Φ (z 0 ) = 0. Hence, Claim 5 together with Proposition 4.3 imply that Let h(z) := z/L + ω, and g(z) := z − ω 2 . Then, the map Ψ := g • Φ • h maps L(J(P ) − ω) onto ∂A f (∞) − ω 2 . Moreover Ψ maps the origin to itself, is C 1+α and has derivative 1 at this point. So, in view of Proposition 4.3, Let us now see that ∂A f (∞) ⊂ J(f ) ⊂ Φ(D). The first inclusion is immediate. If z ∈ J(f ) \ Φ(D), then u := Φ −1 (z) has a bounded forward orbit that avoids D. Since u can never be mapped to S \ D (otherwise z would eventually be mapped to H), we must have Φ −1 (z) ∈ J(P ) ⊂ D. Hence z ∈ Φ(D).
Since To see part (b), notice that ∂S is self-similar by Remark 2.7. The mapping ψ of Theorem 2.6 can be turned into mapping of ∂H 2 by lettingψ = Φ•ψ•Φ −1 . By Propostion 4.1 this composition is C 1+α -conformal or anticonformal, and has the same conformal or anticonformal derivative κ as ψ. We can conclude that ∂H 2 is self-similar in the sense of McMullen.
For part (c), assume that S contains an open triangle T with vertices a, b and ω. Shrinking T we can assume T ∩ ∂S = {ω}. We know from Theorem A (d), that Φ is C-differentiable at ω and Φ (ω) = 0. Moreover, by Proposition 4.1, we have that Φ −1 is also C-differentiable at ω 2 . The image Φ([ω, a]) is a curve in H having a tangent at the starting point, and Φ([ω, b]) is a curve in H having a tangent at the starting point. The angle between the two tangents is the same as the angle of T at ω, since Φ (ω) exists and is non-vanishing. Thus a line segment in the gap between the the two tangents, will not intersect the two image curves in a small enough neighborhood of ω 2 . Hence there is room for a triangle in H with a vertex at ω 2 . As the inverse map Φ −1 is C-differentiable at ω 2 with non-vanishing derivative, we can use the exact same argument to establish that the existence of a triangle in H with a vertex at ω 2 implies the existence of a triangle in S with a vertex at ω.

Proof of Corollary 1.3
It only remains to prove Corollary 1.3. The statements of Theorem A and B hold true if we replace ω 2 with ω 1 . When f is viewed as a mapping of the sphere, ω 1 and ω 2 do not play different roles. If we change coordinates by the map z → 1/z, then f takes the form g(z) = 1/f (1/z) = b −1 az+1 z+a , i.e. we get a mapping of the form covered by Theorems A and B. The change of coordinates interchanges outer and inner boundary of H and ω 1 and ω 2 . More precisely, z → 1/z maps ω 1 for f to ω 2 for g, and ω 2 for f to ω 1 for g. Theorems A and B for g use the change of coordinates to get the approiate statements for ω 1 . In particular, when proving the corollary, we can suppose f m (u) = ω 2 .
Let k ≥ 0 and v be such that P k (v) = ω. Let us first see that there exists an inverse branch of P mapping a neighborhood of ω conformally onto a neighborhood of v. Indeed, the boundary of S is included in the accumulation set of the forward orbit of ω which must be infinite. So P j (v) = ω, for j = 0, 1, . . . , k − 1. Since ω is the only zero of P , this implies in particular that (P k ) (v) = 0, which guarantees the existence of the inverse branch.
The same argument shows that if k ≥ 0 and f k (v) = ω 2 , then there exists an invere branch of f k mapping a neighborhood of ω 2 onto a neighborhood of v.
Part (a). We know that Φ conjugates P to f on C \ D. Hence Φ conjugates P n to f n on C \ ∪ n−1 j=0 (D). That means Φ = f −n • Φ • P n in a neighborhood of ω , for a suitably chosen branch of f −n . Since the property of being C 1+α at a point is clearly preserved by compostion with conformal mappings, we get that f −n • Φ • P n is C 1+α -conformal at ω .
Part (b). Using a suitably chosen branch of P −n and Propostion 4.3, we get Using a suitably chosen branch of f −m in the same way, we get Part (c). There exists an inverse branch f −m that maps ω 2 to u, and maps the part of the boundary of H that lies in some neighborhood of ω 2 onto the part of the boundary of H that lies in some neighborhood of u. Using this inverse branch, we can tranport the mapping φ expressing the self-similarity at ω 2 into a mapφ by conjugating with f −n .
Part (d). Is also proven using that inverse branches are conformal isomorphism. We leave the details to the reader.