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 (up to quasiconformal deformation). 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 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 := formed by those points for which the sequence { f n } is normal in some neighborhood, and its complement, the Julia set which is always non-empty. 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 the Fatou components, map onto one another and are eventually periodic [18]. In contrast to this, the dynamics is chaotic on the Julia set. For background on the dynamics of rational maps we refer for example to [5] and [12].
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 is 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. The boundary of this basin is compact in C and coincides with the Julia set of the polynomial.
Periodic Fatou components of rational maps f are completely classified [7]: a periodic component U is either part of a basin of attraction of an attracting or parabolic cycle, or a rotation domain, which means that some iterate of f | U is conformally conjugate to a rigid rotation by an irrational multiple of 2π , called the rotation number. Rotation domains may be simply connected in which case they are called Siegel disks, or doubly connected in which case they are known as Herman rings. By definition, Herman rings separate the Julia set and have a disconnected boundary.
The dynamics of a rational map is determined, to a large extent, by the orbits of its critical points, i.e., the zeros of its derivative. Indeed, any basin of attraction must contain a critical point [7] and every boundary component of a rotation domain is accumulated by a critical orbit (i.e., the orbit of a critical point) [7,16]. In some cases the relation is even stronger: any rotation domain with rotation number of bounded type (see Sect. 1.1) have Jordan boundaries which actually contain some critical point [21]. In this work we will only consider rotation domains with this property.
Herman rings are undoubtedly the least well known among all possible types of periodic Fatou components of rational maps, one reason being that they are not associated to any periodic point with a certain multiplier, as the other types (basins of attraction or Siegel disks) are. Their closest relatives, Siegel disks, are much better understood, both in terms of conditions for their existence and in terms of the different properties that their boundaries possess. Relevant to our work will be, for instance, the string of geometric results about the fine structure of Siegel disks, proven by McMullen in [11], such as self-similarity of the Siegel disk around the critical point or measurable depth of the critical point in the filled Julia set (see Sect. 2).
But there is a procedure to relate Siegel disks and Herman rings, known as Shishikura's surgery (see Sect. 3 and [16]). Roughly speaking, starting with a map that has a Herman ring H (of a certain rotation number), this construction produces a map having a Siegel disk S (of the same rotation number); at the same time it relates both functions via a quasiconformal map which is a partial conjugacy between them. Intuitively, Shishikura's procedure erases the hole from the Herman ring (and from all its preimages), substituting the dynamics there by a rigid rotation (see Fig. 1). The procedure is reversible, and it therefore ties certain problems on Herman rings (such as their existence for given rotation numbers) to the corresponding problem for Siegel disks.
The quasiconformal map mentioned above opens up a possibility to transfer geometric properties between Siegel disks and their corresponding Herman rings. Some of them are fairly obvious to transport: if the boundary of S is a Jordan curve, so will be both boundary components of H ; or they will all contain a critical point or none will. But other geometric properties are not necessarily preserved by general homeomorphisms or quasiconformal maps.
In this paper we study the extra regularity properties of the quasiconformal map and use them to transfer some of McMullen's results about the fine geometry of Siegel disks to corresponding statements about Herman rings. These are the principal contents of Theorems A and B. In the latter, we additionally conclude that the full boundary of the Herman ring is, surprisingly, tightly similar to that of a Siegel disk, even though one of them is disconnected and the other is not. The concept of tight similarity, introduced here, is stronger than regular similarity. In other words, zooming in around the critical point, the holes of the Herman ring tend to become invisible, until the Siegel disk and the Herman ring become indistinguishable from each other (see Fig. 2).
Our study is done using a model family of rational maps of degree 3, which is one of the simplest that exhibits Herman rings of all rotation numbers of Brjuno type (see Sect. 1.1). It also has the property that Shishikura's surgery relates it to the family of quadratic polynomials used in McMullen's results. However, in the same way that McMullen's properties also hold for quadratic-like mappings (i.e., degree 2 holomorphic branched coverings which map a topological disk properly over itself, see [6]), our theorems also extend to appropriate rational-like maps (see Remark 1.5).

Setup and Statement of Results
Arithmetics plays an important role in the dynamics of rotation domains. It is important to distinguish between three nested classes of irrational numbers. For a 1 , a 2 , . . . ∈ N, we let denote the continued fraction expansion with a 1 , a 2 , . . . as coefficients, and denote the convergents by p n q n = [a 1 , a 2 , . . . , a n ].
See [8] for details. An irrational number θ is a quadratic irrational if the sequence of coefficients 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 those 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 characterized 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 the rigid rotation z → e 2πiα z [4,14,17]. If the map f is globally defined, this neighborhood 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 [20].
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 possesses a Siegel disk S centered at z = 0, because the numbers of bounded type form a subset of the Brjuno numbers.
We will work with the simplest family of rational maps that can possess Herman rings, namely for a, b ∈ C and b = 0. Every f a,b has superattracting fixed points at the origin and at infinity. Additionally, there are two other critical points which we denote by ω 1 and ω 2 .
It is well known [4,16,20] that 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 [3], Buff, Fagella, Geyer, and Henriksen show that for every Brjuno number α, there exists a punctured 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. Note that f a,b is not a Blaschke product unless a is real. But every map f a,b with a Herman ring is quasiconformally conjugate to a Blaschke product (for which a ∈ R and |b| = 1), and the conjugacy can be chosen to be conformal off a small neighborhood of the core curve of the Herman ring and its preimages.
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 simply denote this map by f . Hence  (2), where the boundary of the basin of infinity, A P (∞), coincides with J (P).
In [15], 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 vice versa (see Sect. 3). We will use a special case of his construction in the following theorem. (2) and (3). In the setup above, there exists a P-invariant simply connected domain D compactly contained in S, and a quasiconformal mapping :

Theorem 1.1 Let θ be of bounded type and let P and f be as in
The notion of C 1+α -conformality, due to McMullen, is defined as follows.
Note that this is stronger than C−differentiability 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 1.1 is illustrated in Fig. 1. The fact that the quasiconformal conjugacy provided by Theorem 1.1 is C 1+α at ω implies that the boundary components of H are locally similar to the boundary of S near respective critical points. To make a precise statement, we introduce a notion of similarity that is stronger than Tan Lei's notion of asymptotic similarity introduced in [19]. Let B(c, r ) denote the open ball of center c ∈ C and radius r > 0.

Definition 1.3 (Tight similarity)
We say that two compact sets A, B are tightly similar at z 0 if there exist δ, β > 0 and L > 0 such that When A and B are tightly similar at z 0 , we write , the golden mean. The dynamics of P is shown on 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 shown on 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 1.1, there exists a quasiconformal homeomorphism which is conformal outside the red set.
It is easy to check that tight similarity at z 0 is an equivalence relation on the compact subsets of C. The relation between tight similarity and Hausdorff distance d H between compact sets is as follows. If A and B are two compact sets and we set for some λ > 1 as r → 0. (2) and (3). In the setup above, the following are satisfied.

Main Theorem Let θ be of bounded type and let P and f be as in
(a) There exists a scaling factor L ∈ C \ {0} such that The Main Theorem is illustrated in Fig. 2, where we can see the similarity between J (P), ∂ A f (∞), and J ( f ). Even though J ( f ) and ∂ A f (∞) are topologically very different, as we blow up a neighborhood of ω 2 , the "holes" in the preimages of the Herman ring shrink faster than the blow-up rate.
In the illustration, it seems that the holes in the preimages of the Herman ring get increasingly small relative to the size of the preimage and not only relative to the rate of blow-up. Also, it seems that the holes are located further away from the "roots" of the preimages as we blow up. These experimental observations do not follow from the Main Theorem, and we do not know if they are true though we suspect that this behavior does occur quite generally.
The Main Theorem is useful in the numerical search for mappings in the family 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 1.1 we know 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 all iterated preimages of the critical points: and u ∈ ∂ H , then ∂ H is tightly self-similar around u. Remark 1.5 McMullen's results extend to a much more general class than quadratic polynomials, namely to all quadratic-like maps with a fixed point of derivative e 2πiθ , θ is shown. To the right is illustrated the Julia set of f as in (3), with (a, b) chosen so that θ is as above. Going down, we zoom in around ω and ω 2 , respectively, and see that similarity becomes more and more pronounced being irrational of bounded type (see [11,Theorem 5.1]). 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 cubic 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 , 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 is the critical point. In [13], Petersen showed the following theorem.

Theorem 2.1 The Julia set J (P) is locally connected and has zero Lebesgue measure.
McMullen proved a string of geometric results in [11]. He showed that J (P) has Hausdorff dimension strictly less than two. He also showed that ω is a measurable deep point of a subset of the filled-in Julia set K (P) := C \ A P (∞), a condition that is stronger than being a density point of K (P).  Then ω is a measurable deep point of S .
If φ : C → C is quasiconformal and the support of the Beltrami coefficient μ φ :=∂φ/∂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., [9, Chapter 6]), and we will find use for the following theorem of McMullen. McMullen also showed that when θ is a quadratic irrational, the Siegel disk is self-similar at the critical point ω. More precisely he showed the following.  (2) and S be its Siegel disk. Then, there exists a homeomorphism ψ defined in some neighborhood of ω with ψ(ω) = ω, conjugating P q n to P q n+s on ∂ S for n sufficiently large. Moreover,

]) Suppose θ is a quadratic irrational, and let s denote the eventual period of the coefficients of its continued fraction. Let P be as in
if s odd for some α > 0 and κ ∈ C 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, then and, when s is odd, where κ is the scaling factor in Theorem 2.6.
Using the preceding theorem, Buff and Henriksen [2] proved that for some values of θ , such as the golden mean In this section we prove Theorem 1.1. We shall see that parts (a), (b), and (c) follow directly from a surgery construction due to Shishikura, whereas part (d) can be derived by bounding the relative area of the support of the quasiconformal distortion of 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 denote the conjugacy from P to F, we then check that it has the stated properties. Details are as follows (c.f. [16] and [1, Sect. 7.2]). We keep the notation from the setup in Sect. 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 complement of the unbounded component of C \ 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 Fig. 3.
We modify and extend φ 0 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 . This way we get a quasiconformal mapping φ 1 : U 3 → V 3 which we shall use to paste the rigid rotation into the Herman ring. Define Note that the two maps agree on ∂U 2 so F is continuous. 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. We start by defining μ in U 3 , by pulling back the standard Beltrami coefficient μ 0 = 0 under φ 1 , that is μ = φ * 1 (μ 0 ) on U 3 or equivalently, μ =∂φ 1 /∂φ 1 . Observe that μ is invariant by F| U 3 by the construction, and it has bounded dilatation because φ 1 is quasiconformal. We can extend μ 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 = μ 0 outside ∞ n=0 F −n (U 3 ). Since μ on U 3 is invariant by F| U 3 , the extended μ is invariant by F. Also, since F is analytic outside U 2 it follows that ||μ|| ∞ = ||μ | U 3 || ∞ < 1. Hence, F is holomorphic with respect to the almost complex structure defined by μ, and therefore we can apply the measurable Riemann mapping theorem (see e.g., [1,Theorem 1.27]) to obtain a quasiconformal homeomorphism : C → C, satisfying μ = * (μ 0 ) or, equivalently,∂ = μ ∂ . Since is unique up to fixing the image of three points, we can normalize it by requiring (ω 2 ) = ω = −e 2πiθ /2, (0) = 0, and (∞) = ∞.
The map Q = • F • −1 is a quadratic polynomial. Indeed, it is a rational map of degree two with Q −1 (∞) = ∞. Observe that 0 is fixed by Q. Since φ 1 is conformal on U 1 , so is F, and F (0) = R θ (0) = e 2πiθ . This implies that is also conformal on U 1 and hence Q (0) = F (0) = e 2πiθ . Hence z = 0 is a Siegel point of Q. Additionally, ω is a critical point, thus we conclude that Q = P.
Property (a) First notice that φ 1 • is conformal on (U 3 ) and conjugates P to R θ on this domain.
were only a subdisk of S then H would not be the maximal domain of linearization. By construction, conjugates P to F everywhere, but since F = f except on U 2 , conjugates P to f everywhere except on D. Finally, since U 2 is compactly contained in U 3 , we know D is compactly contained in S.
conjugates P to f . : ω is a measurable deep point of 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 The preimage P −1 (D) has two components: D itself and another one D . If z ∈ D \ D, some forward iterate of z must hit D . Hence z does not belong to S if is less than the distance between D and D . This proves the desired inclusion and concludes the proof of Theorem 1.1.
In the course of the proof, we showed that ω is a measurable deep point of C \ D. We shall use this later, so we formally state it.

Preliminaries About C 1+α -Conformal Mappings and Tightly Similar Sets
To prove the Main Theorem, we need to establish some elementary properties of C 1+α -conformal homeomorphisms and tightly similar sets. First we prove that C 1+α -conformality extends to inverses.
The following Proposition is straightforward and left as an exercise for the reader.

Proposition 4.2 Let A ⊂ B ⊂ C be compact sets. Then
We end this section by showing that tight similarity at a point is preserved by homeomorphisms which are C 1+α -conformal at that point.

If φ (z 0 ) = 0 and B is a compact such that A
Proof To see the first statement, first write φ(z) = z + R(z), where |R(z)| ≤ M|z − z 0 | 1+α for |z| small. Then, for any a ∈ A ∩ B(z 0 , δ), This is the first requirement of Definition 1.3. 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 simultaneously scaling A and B around z 0 by the same factor keeps them tightly similar, we can suppose φ (z 0 ) = 1. Then so 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 eventual period of the continued fraction expansion of θ , and first assume it is even. Let ψ be the homeomorphism and κ the scaling factor in Theorem 2.6, which ensure self-similarity of S. Then, the map In this section we prove the Main Theorem. Most of the statements are fairly straightforward by the results we have established so far. The delicate part is to prove that Recall that denotes the quasiconformal homeomorphism given in Theorem 1.1, 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) ∼ where α denotes the "center" of D , i.e., the unique point in D whose forward orbit eventually hits the fixed point 0.
The claim follows from Koebe's Distortion Theorem (see e.g., [5]) . Let φ : S → B(0, 1) denote a map linearizing P, and F be the Claim 2 There exists K 2 , K 3 ∈ (0, ∞) such that for any connected component D of D, where α is the center of D .
There are only finitely many components D of D withr (D , α ) > r 0 . If there were infinitely many, then by Claim 2, the area of D would be infinite, but D is contained in the filled-in Julia set of P and thus has finite area. 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| + 2r < δ + 2r 0 < δ, we obtain Using Claim 2, we get Since r ≤ r 0 , and by the choice of r 0 , the left-hand side is bounded below by r γ 2 , and we obtain r ≤ K 5 |z − ω| 1+β/2 , for some K 5 ∈ (0, ∞). By Claim 3, d(z, J (P)) ≤ K 4 K 5 |z − ω| 1+β/2 which proves Claim 4.
We have just established one of the two requirements of Definition 1.3. The set D is a completely invariant closed set containing more than two points, so J (P) ⊂ D. Hence, the other requirement is automatically satisfied.
Proof of the Main Theorem. By Theorem 1.1, (J (P)) = ∂ A f (∞). Again using Theorem 1.1 we obtain that L := (ω) = 0. Let h(z) := z/L +ω, and g(z) := z−ω 2 . Then, the map Moreover maps the origin to itself, is C 1+α -conformal and has derivative 1 at this point. By Proposition 4.3, Let us now see that ∂ A f (∞) ⊂ J ( f ) ⊂ (D). The first inclusion is immediate. If z ∈ J ( f ) \ (D), then u := −1 (z) has a bounded forward orbit that avoids D. Since the orbit of u cannot hit S \ D (otherwise the orbit of z would hit H ), we must have −1 (z) ∈ J (P) ⊂ D. Hence z ∈ (D).
By claim 5 and Proposition 4.
This finishes the proof of part (a).
To see part (b), notice that ∂ S is self-similar by Remark 4.4. The mapping ψ of Theorem 2.6 can be turned into mapping of ∂ 2 H by lettingψ = • ψ • −1 . By Proposition 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 ∂ 2 H admits a map such as the one in Theorem 2.6, and by Remark 4.4 is tightly self-similar.
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 1.1 (d) that is C-differentiable at ω and (ω) = 0. Moreover, by Proposition 4.1, we have that −1 is also C-differentiable at ω 2 . The image ([ω, a]) is a curve in H having a tangent at the starting point, and similarly for ([ω, b]). The angle between the two tangents is the same as the angle of T at ω, since (ω) exists and is non-vanishing. Thus a line segment in the gap between 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 −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. Remark 5.2 The proof shows that ∂ 2 H admits a map such as the one in Theorem 2.6, which is stronger than part (b).
It only remains to prove Corollary 1.4. The statements of Theorem 1.1 and the Main Theorem 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 z 2 az+1 z+a , i.e., we get a mapping of the form covered by Theorem 1.1 and the Main Theorem. The change of coordinates interchanges outer and inner boundary components of H and maps ω 1 for f to ω 2 for g, and ω 2 for f to ω 1 for g. In particular, in proving the corollary, we can suppose f m (u) = ω 2 .
Let k ≥ 1 and v be such that P k (v) = ω. There exists an inverse branch of P mapping a neighborhood of ω conformally onto a neighborhood of v. Indeed, the boundary of S is contained 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 local inverse branch.
The same argument shows that if k ≥ 1 and f k (u) = ω 2 , then there exists an inverse branch of f k mapping a neighborhood of ω 2 onto a neighborhood of u.
Proof of Corollary 1.4. Part (a) We know that conjugates P to f on C \ D. Hence conjugates P n to f n on C \ n−1 j=0 P − j (D). That means = f −n • • 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 under composition with conformal mappings, we conclude that is C 1+α -conformal at ω . Part (b) Using a suitably chosen branch of P −n and Proposition 4.3, we get with L 1 = (P n ) (ω ). Using a suitably chosen branch of f −m in the same way, we get with L 2 = ( f m ) (u). It follows from the Main Theorem that (b) holds with L = L 1 (ω)L −1 2 . 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. If φ is the map expressing the self-similarity at ω 2 , then f −n • φ • f n will be a map that expresses self-similarity of ∂ H at u.