The high type quadratic Siegel disks are Jordan domains

Let $\alpha$ be an irrational number of sufficiently high type and suppose $P_\alpha(z)=e^{2\pi i\alpha}z+z^2$ has a Siegel disk $\Delta_\alpha$ centered at the origin. We prove that the boundary of $\Delta_\alpha$ is a Jordan curve, and that it contains the critical point $-e^{2\pi i\alpha}/2$ if and only if $\alpha$ is a Herman number.

Let f be a non-linear holomorphic function with f (0) = 0 and f ′ (0) = e 2πiα , where 0 < α < 1 is an irrational number.We say that f is locally linearizable at the fixed point 0 if there exists a holomorphic function defined near 0 which conjugates f to the rigid rotation R α (z) = e 2πiα z.The maximal region in which f is conjugate to R α is a simply connected domain called the Siegel disk of f centered at 0.
The existence of the Siegel disk of f is dependent on the arithmetic condition of α ∈ (0, 1) be the continued fraction expansion of α.The rational numbers p n /q n := [0; a 1 , • • • , a n ], n ⩾ 1, are the convergents of α, where p n and q n are coprime positive integers.If α belongs to the Brjuno class n log q n+1 < +∞}, then any holomorphic germ f with f (0) = 0 and f ′ (0) = e 2πiα is locally linearizable at 0 and hence f has a Siegel disk centered at the origin [Sie42,Brj71].Yoccoz proved that the Brjuno condition is also necessary for the local linearization of the quadratic polynomial P α (z) := e 2πiα z + z 2 : C → C at the origin [Yoc95].
1.1.Topology and obstructions of Siegel disk boundaries.The dynamics in the Siegel disks is simple and one mainly concerns the properties on the boundaries.In the 1980s, Douady and Sullivan asked the following question (see [Dou83], [Rog92a]): Question.Is the boundary of a Siegel disk a Jordan curve?
An important breakthrough was made by Petersen and Zakeri in 2004.They proved that for almost all irrational number α, the boundary of the Siegel disk of the quadratic polynomial P α is a Jordan curve [PZ04].We refer these irrational numbers the PZ type, i.e., log a n = O( √ n) as n → ∞, where a n is the n-th digit of the continued fraction expansion of α.Recently, Zhang generalized this result to all polynomials [Zha14] and obtained the same result for the sine family [Zha16].
Suppose the closure of the Siegel disk of f is compactly contained in the domain of definition of f .One may wonder what phenomena near the boundary of a Siegel disk prevents f from having a larger linearization domain.Obviously, the presence of periodic cycles near the boundary is one of the reasons since any Siegel disk cannot contain periodic points except the center itself.It was proved by Avila and Cheraghi that under some condition on α every neighborhood of the Siegel disk of P α contains infinitely many cycles [AC18], which is similar to the small cycle property that prevents linearization (see [Yoc88] and [Pér92]).
On the other hand, note that any Siegel disk cannot contain a critical point.Hence the second question on the Siegel disk boundary is: Does the boundary of a Siegel disk always contain a critical point?The answer is no.Ghys and Herman gave the first examples of polynomials having a Siegel disk whose boundary does not contain a critical point (see [Ghy84], [Her86] and [Dou87]).
In relation to the results on the regularity1 of the boundaries of the Siegel disks mentioned above (for the bounded type or PZ type rotation numbers), they also include the statement that the boundaries of those Siegel disks pass through at least one critical point.In particular, for the bounded type rotation numbers, Graczyk and Światek proved a very general result: if an analytic function has a Siegel disk properly contained in the domain of holomorphy and the rotation number is of bounded type, then the boundary of the corresponding Siegel disk contains a critical point [G Ś03].
Herman was one of the pioneers who studied the analytic diffeomorphisms on the circles [Her79].He introduced the following subset of irrational numbers.
Definition (Herman numbers).Let H be the set of irrational numbers α such that every orientation-preserving analytic circle diffeomorphism of rotation number α is analytically conjugate to the rigid rotation.
Herman proved that the set H is non-empty and contains a subset of Diophantine numbers [Her79].Yoccoz proved that H contains all Diophantine numbers (and hence contains all bounded type and PZ type numbers), and also gave an arithmetic characterization of the numbers in H [Yoc02].
Suppose f is an analytic function which has a Siegel disk properly contained in the domain of holomorphy.Ghys proved that if the rotation number belongs to H and the boundary of the Siegel disk is a Jordan curve, then f has a critical point in the boundary of the Siegel disk [Ghy84].Later, Herman generalized this result by dropping the topological condition on the Siegel disk boundary but requiring that the restriction of f on the Siegel disk boundary is injective [Her85] (see also [Pér97]).In particular, he proved that if a unicritical polynomial has a Siegel disk whose rotation number is contained in H, then the boundary of the Siegel disk contains a critical point.Recently, Chéritat and Roesch, Benini and Fagella, respectively, generalized this result to the polynomials with two critical values [CR16] and to a special class of transcendental entire functions with two singular values [BF18].
For polynomials, Rogers proved that if the Siegel disk ∆ is fixed and the rotation number is in H, then either ∂∆ contains a critical point or ∂∆ is an indecomposable continuum [Rog98].For the exponential map E θ (z) = e 2πiθ (e z − 1), it was proved by Herman that, if E θ has a bounded Siegel disk ∆ θ , then E θ is injective on ∂∆ θ .Hence it follows from Herman's result that ∆ θ is unbounded when θ ∈ H since E θ has no critical points [Her85].Conversely, Herman, Baker and Rippon asked a question: if ∆ θ is unbounded, is necessarily the singular value −e 2πiθ contained in ∂∆ θ ?Rippon showed that this is true for almost all θ [Rip94] and the question was fully answered positively by Rempe [Rem04] and independently by Buff and Fagella (unpublished).Moreover, Rempe also studied the Herman type Siegel disks of some other transcendental entire functions [Rem08].
1.2.The statement of the main result.The proofs of the regularity results for the bounded type and PZ type Siegel disks stated previously are all based on surgery: either quasiconformal or trans-quasiconformal.In these proofs, some premodels, and usually, a single or a family of Blaschke products are needed.By surgery, the regularity and the existence of critical points on the boundaries of Siegel disks were proved at the same time.
In this paper, without using surgeries we shall prove that the Siegel disks of some holomorphic maps are Jordan domains and that Herman type rotation number is also necessary for the existence of critical points on the Siegel disk boundaries.To this end, it requires us to restrict the rotation numbers to a special class since we need to use near-parabolic renormalization scheme.In [IS08], a renormalization operator R and a compact class F that is invariant under R were introduced.All the maps in F have a special covering structure.They have a neutral fixed point at the origin and a unique simple critical point in their domains of definition.The renormalization operator assigns a new map in F to a given map of F that is obtained by considering the return map to a sector landing at the origin.As a return map, one iterate of Rf corresponds to many iterates of f ∈ F. To study very large iterates of f near 0, one hopes to repeat this process infinitely many times.However, to iterate R infinitely many times, the scheme requires the rotation number α, where f ′ (0) = e 2πiα , to be of high type, that is, α belongs to for some big integer2 N ∈ N. In this paper we prove the following main result.
Main Theorem.Let α be an irrational number of sufficiently high type and suppose P α (z) = e 2πiα z + z 2 has a Siegel disk ∆ α centered at the origin.Then the boundary of ∆ α is a Jordan curve.Moreover, it contains the critical point −e 2πiα /2 if and only if α is a Herman number.
Note that HT N has measure zero if N ⩾ 2. However, all the usual types of irrational numbers have non-empty intersections with HT N : bounded type, PZ type, Herman type and Brjuno type etc.In particular, HT N contains some irrational numbers such that the Siegel disk boundary of P α has the regularity studied in [ABC04], [BC07] and the self-similarity studied in [McM98].Rogers proved that the boundary of any bounded irreducible Siegel disk ∆ is either tame: the conformal map from ∆ to the unit disk has a continuous extension to ∂∆, or wild: ∂∆ is an indecomposable continuum [Rog92b].Recently, Chéritat constructed a holomorphic germ such that the corresponding Siegel disk is compactly contained in the domain of definition but the boundary is not locally connected [Ché11].Our main theorem indicates that the boundaries of quadratic Siegel disks should be tame.
As we have seen, in order to guarantee the existence of critical points on the boundaries of Siegel disks, Herman condition (i.e., the rotation number is of Herman type) appears usually as a requirement of sufficiency in most of the literature.As far as we know, the necessity only appears in [BCR09], where it proves that Herman condition is equivalent to the existence of a critical point on the boundary of the Siegel disks of a family of toy models.
In fact, besides the quadratic polynomials, the proof of the Main Theorem in this paper is also valid for all the maps in Inou-Shishikura's invariant class.Hence the Main Theorem is also true for some rational maps and transcendental entire functions.We would like to point out that it was proved in [Yam08] and [AL22] that the bounded type Siegel disks of the maps in Inou-Shishikura's class are quasi-disks if the rotation number is of sufficiently high type.
By constructing topological models of the post-critical sets of the maps in the Inou-Shishikura's class for all high type numbers, Cheraghi gave an alternative proof of the Main Theorem independently (see [Che22a]).Our proofs are different: we analyze the dynamics and carry out the computations in the renormalization tower directly.
Recently, Dudko and Lyubich made significant progress on the quadratic Siegel polynomials P α [DL22].They proved that the restriction of P α on the boundary of the Siegel disk ∆ α of P α is injective, which implies that ∂∆ α is not the whole Julia set of P α (actually they proved a more general result for all α ∈ R \ Q).
1.3.Strategy of the proof.Let f 0 be the normalized quadratic polynomial or a map in Inou-Shishikura's class (see §2.1) satisfying f 0 (0) = 0 and f ′ 0 (0) = e 2πiα , where α is of Brjuno type and of sufficiently high type.For n ⩾ 0, let f n+1 = Rf n be the sequence of the maps which are generated by the near-parabolic renormalization operator R. For each n ⩾ 0, we use P n to denote the perturbed petal of f n and Φ n the corresponding perturbed Fatou coordinate (see definitions in §2.2).
In order to prove that the boundary of the Siegel disk of f 0 is a Jordan curve, we construct a sequence of continuous curves (γ n 0 : [0, 1] → C) n∈N in the perturbed Fatou coordinate plane of f 0 by using a renormalization tower.Each γ n 0 is obtained from γ 0 n (in the perturbed Fatou coordinate plane of f n ) by going up through the renormalization tower, i.e., by lifting and then spreading around.In Lemma 3.2 we show that the inner radius of the Siegel disk ∆ n of f n is estimated by the Brjuno sum up to a multiplicative constant.Then we choose the suitable height of γ The key ingredient is Proposition 4.5: the sequence of the continuous curves (γ n 0 : [0, 1] → C) n∈N converges uniformly to a limit γ ∞ : [0, 1] → C, which is also a continuous curve.For the proof, we use a family of "straight" curves η 0 n to encode the difference between γ 0 n and γ 1 n in the Fatou coordinate plane of f n .The diameters of the η 0 n are discussed in Step 2 of the proof.The diameters of the lifts of η 0 n are estimated by two kinds of contraction: one is the uniform contraction with respect to the hyperbolic metrics in subdomains of the renormalization tower (see Lemma 4.7) and the other is "Brjuno-type arithmetic" -estimates from §2.4 (see also Lemma 4.8).In conclusion, the oscillations of the curves (γ n 0 : [0, 1] → C) n∈N are bounded in terms of the Brjuno sum, i.e., (γ n 0 : [0, 1] → C) n∈N form an equicontinuous family.Because of the contraction by going up the renormalization tower, the sequence Φ −1 0 (γ n 0 ) converges exponentially fast towards the boundary of ∆ 0 (see Proposition 4.9).
For the second part of the Main Theorem which concerns Herman condition, we construct a Jordan arc Γ 0 in the non-escaping set of f 0 which connects the unique critical value cv with the origin, where γ 0 := Φ 0 (Γ 0 ) is contained in a half-infinite strip 0 with finite width.The existence of Γ 0 is proved in Lemma 5.3 and the proof is also based on the contraction via going up the renormalization tower.To apply the contraction property successfully, the shape of Φ −1 0 (0) needs to be controlled and this is Lemma 5.1 whose proof is given in the Appendix.The construction of Γ 0 guarantees that Γ n = Exp • Φ n−1 (Γ n−1 ) is also a Jordan arc connecting cv with the origin and γ n = Φ n (Γ n ) is contained in 0 for all n ⩾ 1.
We study the homeomorphism s αn := Φ n • Exp : γ n−1 → γ n from the simple curve in one level of the renormalization to another.Lemmas 5.4 and 5.5 estimate the dynamics of the s αn in terms of the Brjuno sum.Based on the sequence (s αn ) n∈N , we define a new class of irrational numbers H N which is a subset of Brjuno numbers, where N is a large number.After comparing the properties of s αn and Yoccoz's arithmetic characterization of H, we prove that H N is exactly equal to the set of high type Herman numbers (see Lemmas 6.4 and 6.6).On the other hand, we prove that the boundary of the Siegel disk of f 0 contains the critical value cv if and only if α ∈ H N (see Proposition 5.7).This implies that the second part of the Main Theorem holds.
1.4.Some observations.There are several applications of Inou-Shishikura's invariant class.The first remarkable application is that Buff and Chéritat used it as one of the main tools to prove the existence of Julia sets of quadratic polynomials with positive area [BC12].Recently, Cheraghi and his collaborators have found several other important applications.In [Che13] and [Che19], Cheraghi developed several elaborate analytic techniques based on Inou-Shishikura's results.The tools in [Che13] and [Che19] have led to part of the recent major progresses on the dynamics of quadratic polynomials.For examples, the Feigenbaum Julia sets with positive area (which is different from the examples in [BC12]) have been found in [AL22], the Marmi-Moussa-Yoccoz conjecture for rotation numbers of high type has been proved in [CC15], the local connectivity of the Mandelbrot set at some infinitely satellite renormalizable points was proved in [CS15], some statistical properties of the quadratic polynomials was depicted in [AC18], the topological structure and the Hausdorff dimension of high type irrationally indifferent attractors were characterized in [Che22a] and [CDY20] respectively.
Recently, Chéritat generalized the near-parabolic renormalization theory to the unicritical families of any finite degrees [Ché22b].See also [Yan21] for the corresponding theory of local degree three.Hence there is a hope to generalize the Main Theorem in this paper to all unicritical polynomials.
Acknowledgements.We would like to thank Xavier Buff and Arnaud Chéritat for helpful discussions and offering the manuscript [BCR09].We are also very grateful to Davoud Cheraghi for pointing out a gap in an earlier version of the paper and providing many invaluable comments and suggestions.The second author was indebted to Institut de Mathématiques de Toulouse for its hospitality during his visit in 2014/2015 where and when partial of this paper was written.He would also like to thank Davoud Cheraghi and Arnaud Chéritat for persistent encouragements.We are very grateful to both referees for their very insightful comments and suggestions.This work was supported by NSFC (grant Nos.12222107, 12071210), NSF of Jiangsu Province (grant No. BK20191246) and the CSC program (2014/2015).
Notations.We use N, N + , Z, Q, R and C to denote the set of all natural numbers, positive integers, integers, rational numbers, real numbers and complex numbers, respectively.The Riemann sphere and the unit disk are denoted by C = C ∪ {∞} and D = {z ∈ C : |z| < 1} respectively.A round disk in C is denoted by D(a, r) = {z ∈ C : |z −a| < r} and D(a, r) is its closure.Let x ∈ R be a non-negative number, we use ⌊x⌋ to denote the integer part of x.
For a set X ⊂ C and a number δ > 0, let B δ (X) := z∈X D(z, δ) be the δneighborhood of X.For a number a ∈ C and a set X ⊂ C, we denote aX := {az : z ∈ X} and X ± a := {z ± a : z ∈ X}.Let A, B be two subsets in C. We say that A is compactly contained in B if the closure of A is compact and contained in the interior int(B) of B and we denote it by A ⋐ B. We use diam(X) to denote the Euclidean diameter of a set X ⊂ C and len(γ) the Euclidean length of a rectifiable curve γ ⊂ C.

Near-parabolic renormalization scheme
In this section, we summarize some results in [IS08], [BC12], [AC18] and [Che19] which will be used in this paper.Parts of the theories can be also found in [Shi98] and [Shi00].
2.1.Inou-Shishikura's class.Let P (z) := z(1+z) 2 be a cubic polynomial with a parabolic fixed point at 0 with multiplier 1.Then P has a critical point cp P := −1/3 which is mapped to the critical value cv P := −4/27.It has also another critical point −1 which is mapped to 0. Consider the ellipse and define 3 (2.2) The domain U is symmetric about the real axis, contains the parabolic fixed point 0 and the critical point cp P , but U ∩ (−∞, −1] = ∅ (see [IS08, §5.A] and Figure 1).
3 The domain U is denoted by V in [IS08].For a given function f , we denote its domain of definition by U f .Following [IS08, §4], we define a class of maps4 Each map in this class has a parabolic fixed point at 0, a unique critical point at cp f := φ(−1/3) ∈ U f and a unique critical value at cv := −4/27 which is independent of f .For α ∈ R, we define For convenience, we normalize the quadratic polynomials to such that all Q α have the same critical value −4/27 as the maps in IS α .In particular, We would like to mention that the quadratic polynomial Q α is not in the class IS α . Theorem Normalization of Φ attr,f and Φ rep,f .The univalent map Φ attr,f (resp.Φ rep,f ) in Theorem 2.1 is called an attracting (resp.repelling) Fatou coordinate of f and P attr,f (resp.P rep,f ) is called an attracting (resp.repelling) petal.The attracting Fatou coordinate Φ attr,f can be naturally extended to the immediate attracting basin A attr,f of 0. Specifically, for z ∈ A attr,f such that f •k (z) ∈ P attr,f with k ⩾ 0, one can define Since Φ attr,f is unique up to an additive constant, we normalize it by Φ attr,f (cp f ) = 0. Therefore, we have Φ attr,f (P attr,f ) = {ζ ∈ C : Re ζ > 0}.
Every f ∈ IS 0 ∪ {Q 0 } can be written as where γ = 1 − a 3 /a 2 2 is the iterative residue of f and C attr , C rep are constants.Since Φ rep,f is also unique up to an additive constant, we normalize it by setting 2.2.Near-parabolic renormalization.We need to consider the case that a sequence of functions converges to a limiting function and the neighborhoods of a function need to be defined.

Definition (Neighborhoods of a function). Let
where d C denotes the spherical distance, K is a compact subset contained in U f and ε > 0. A sequence (f n ) is said to converge to f uniformly on compact sets if for any neighborhood N of f , there exists n 0 > 0 such that f n ∈ N for all n ⩾ n 0 .
Proposition 2.2 ([BC12, Proposition 12, p. 707], see Figure 2).There exist k ∈ N + and ε 1 > 0 satisfying ⌊ 1 ε1 ⌋−k > 1, such that for all f ∈ IS α ∪{Q α } with α ∈ (0, ε 1 ], there exist a Jordan domain P f ⊂ U f and a univalent map Φ f : P f → C, such that (a) P f contains cv and it is bounded by two arcs joining 0 and σ f ;    Proposition 2.2 was proved in [BC12] only for Inou-Shishikura's class.However, when f = Q α with sufficiently small α > 0, the existence of the domain P f and the coordinate Φ f : P f → C satisfying the properties in the above proposition is classic (see [Shi00]).The map Φ f in Proposition 2.2 is called the (perturbed) Fatou coordinate of f and P f is called a (perturbed) petal.
Proposition 2.3 ([Che19, Proposition 2.7], see Figure 3).There exist constants is an isomorphism, and the unique connected component Definition (Near-parabolic renormalization, see Figure 3  to a well-defined map Rf which is defined on a set punctured at zero, where s : z → z is the complex conjugacy.One can check that Rf extends across zero and satisfies (Rf )(0) = 0 and (Rf ) ′ (0) = e 2πi/α f .The map Rf is called the nearparabolic renormalization6 of f .
, the near-parabolic renormalization Rf is well-defined and the restriction of Rf in a domain containing 0 can be written as P • ψ −1 ∈ IS 1/α .Moreover, ψ extends to a univalent function on e −2πi/α U ′ .
From Theorem 2.4 we know that the near-parabolic renormalization of Rf can be also defined if the fractional part of 1/α is contained in (0, ε ′ 1 ].This implies that the near-parabolic renormalization operator R can be applied infinitely many times to f if α is of sufficiently high type.
2.3.Some sets in the Fatou coordinate planes.For a set X in C, we use int(X) to denote the interior of X.Let f ∈ IS α ∪ {Q α } with α ∈ (0, ε ′ 1 ].We define a set in the Fatou coordinate plane of f : (2.6) Lemma 2.5.The map Φ −1 f : Φ f (P f ) → P f can be extended to a holomorphic map . This lemma has been proved in [AC18, Lemma 1.8].For completeness and clarifying some ideas we include a sketch of the construction of Φ −1 f here.Proof.By (2.3), the definition of S f , Propositions 2.2(b) and 2.3(a), we have ).Note that there may exist two choices8 of j for some point ζ.Assume that ζ ∈ Φ f (S f ) + j ′ for some j ′ ∈ [1, b f ] and j ′ ̸ = j.Then |j ′ − j| = 1.Without loss of generality, we assume that j ′ = j + 1.By Proposition 2.2(c), we have The modified exponential map Exp : Proposition 2.6 ([AC18, Proposition 1.9]).There exists Proposition 2.6 was proved by applying Proposition 2.3, the pre-compactness of the class IS α and a uniform bound on the total spiral of the set P f about the origin (see [BC12,Proposition 12] or [Che19, Proposition 2.4]).
Lemma 2.8.There are constants , and for all r ∈ (0, 1/2] and all The following Lemma 2.9 and Proposition 2.10 are useful in the estimates of the locations of the points under Φ −1 f and χ f .Lemma 2.9.There exists a constant D 0 > 0 such that for any D ′ 1 > 0, there exists By the continuous dependence of the Fatou coordinates of the maps in IS 0 , the pre-compactness of IS 0 and note that P f is compactly contained in the domain of definition of f , there exists a constant R 1 > 0 such that By (2.12) and the formula of τ f in (2.13), a direct calculation shows that there exists a constant Without loss of generality we assume that D ′ 1 > 1. Combining (2.15) and (2.16), there exists a constant C 2 > 0 depending only on On the other hand, by Lemma 2.8(a) and applying By Lemma 2.8(d) and [Che19, Proposition 6.16], there exists a constant Proposition 2.10 ([Che19, Propositions 6.19 and 6.17]).There are constants However, the arguments there can be applied to completely similarly by using [Che19, Lemma 6.7] and Lemma 2.9.For more details on the study of L f and L −1 f , see [Che19, § §6.3-6.6] and [CS15, §3.5].
Let X ⩾ 0 and Y ⩾ 0 be two numbers.We use X ≍ Y to denote that X and Y are in the same order, i.e., there exist two universal positive constants C 1 and C 2 such that C 1 Y ⩽ X ⩽ C 2 Y .Let D f be the set defined in (2.9).
Lemma 2.11.There exist constants ε 3 ∈ (0, ε ′ 2 ] and D 3 > 0 such that for all . By the pre-compactness of IS α , it is sufficient to prove the statements in this lemma for ζ ∈ Φ f (P f ).

and we have (2.22).
Summering the constants in Parts (a) and (b), the lemma follows if we set D 4 := D 3 + 1 and D 5 := log(4 + 2D 3 ) + 6πD 3 + 2π.□ In the following, we use h ′ to denote ∂h/∂z if h is holomorphic and denote ∂h/∂z if h is anti-holomorphic.The following result is useful in the estimate of the Euclidean length of curves in Fatou coordinate planes.Proposition 2.13.There exist positive constants where , by a direct calculation 12 we have min (2.23) . By (2.4), (2.13), (2.14) and a straightforward calculation we have .
Let α ∈ HT N and f 0 ∈ IS α ∪ {Q α }.By Theorem 2.4, the following sequence of maps is well-defined for all n ⩾ 0: Let U n := U fn be the domain of definition of f n for n ⩾ 0. Then for all n, we have and cv = cv fn = −4/27.For n ⩾ 0, let Φ n := Φ fn be the Fatou coordinate of f n : U n → C defined in the perturbed petal P n := P fn and let C n := C fn and C ♯ n := C ♯ fn be the corresponding sets for f n defined in (2.3).Let k n := k fn be the positive integer in Proposition 2.3 such that For n ⩾ 0, let D n := D fn and D n := D fn be the sets defined in (2.6) and (2.9) respectively.Note that D n ⊂ D n by Lemma 2.7.According to Lemma 2.5, we have a holomorphic map We denote the lift χ fn,0 in (2.8) by χ n,0 .Then, for n ⩾ 1 we have Each χ n,0 is anti-holomorphic.For j ∈ Z we define χ n,j := χ n,0 + j. (2.37) In the following we are mainly interested in χ n,j with 0 ⩽ j ⩽ a n = ⌊ 1 αn−1 ⌋.For δ > 0, let B δ (X) be the δ-neighborhood of a set X ⊂ C with respect to the Euclidean metric.The following lemma will be used to prove the uniform contraction with respect to the hyperbolic metrics in the domains of adjacent renormalization levels (see Lemma 4.7).

Lemma 2.14 ([AC18, Lemma 2.1]
).There exists a constant δ 0 > 0 depending only on the class IS 0 , such that for all n ⩾ 1 and 0 ⩽ j ⩽ a n , then For n ⩾ 0, recall that P n is the perturbed petal of f n .For n ⩾ 1, we define an anti-holomorphic map ψ n by (2.38) Hence we have the following diagrams: Each ψ n extends continuously to 0 ∈ ∂P n by mapping it to 0. For n ⩾ 1, we define the composition For n ⩾ 0 and i ⩾ 1, define the sector In particular, S n 0 ⊂ P 0 for all n ⩾ 0. Define ; and (c) For every m < n, f n : P ′ n → P n and f •kn In particular, the dynamics of f n is conjugate to the dynamics of f 0 .Specifically, the first k n iterates of f n on S 0 n corresponds to k n q n + q n−1 iterates of f 0 and the next From the definition of D n in (2.6) and by Lemma 2.15, the following sets are welldefined for each n ⩾ 0 : Definition (High type Brjunos).Let N be the integer fixed before.Define , where ∆ 0 is the Siegel disk of f 0 .In the rest of this paper, unless otherwise stated, for a given map f 0 ∈ IS α ∪ {Q α } with α ∈ HT N , we use f n to denote the map after n-th near-parabolic renormalization.We also use U n , P n and Φ n etc to denote the domain of definition, the perturbed petal and the Fatou coordinate etc of f n respectively.Theorem 3.1 (Koebe's distortion theorem).Suppose f : D → C is a univalent map with f (0) = 0 and f ′ (0) = 1.Then for each z ∈ D we have and α n ∈ (0, 1) be the number defined inductively as in (2.35) for n ⩾ 1. Denote β −1 = 1 and β n := n i=0 α i for n ⩾ 0. The Brjuno sum B(α) of α in the sense of Yoccoz is defined as Suppose a holomorphic map f has a Siegel disk ∆ f centered at the origin which is compactly contained in the domain of definition of f .The inner radius of ∆ f is the radius of the largest open disk centered at the origin that is contained in ∆ f .Lemma 3.2.There exists a universal constant D 7 > 1 such that for all f 0 ∈ IS α ∪ {Q α } with α ∈ B N , the inner radius of the Siegel disk of f n is c n e −B(αn) with 1/D 7 ⩽ c n ⩽ D 7 for every n ∈ N.
Proof.By the definition of near-parabolic renormalization, it follows that f n ∈ IS αn with α n ∈ B N for all n ⩾ 1.Then according to [Brj71], each f n with n ⩾ 0 has a Siegel disk centered at the origin.By the definition of Inou-Shishikura's class and Koebe's distortion theorem (Theorem 3.1(b)), f n is univalent in D(0, c) for a universal constant c > 0. According to Yoccoz [Yoc95, p. 21], the Siegel disk of f n contains a round disk D(0, C 1 e −B(αn) ) for a universal constant C 1 > 0, where is the Brjuno sum of α n defined in (3.1).On the other hand, by [Che19, Theorem G], there is a universal constant C 2 > 1 such that the inner radius of the Siegel disk of f n is bounded above by C 2 e −B(αn) for all n ∈ N. The lemma follows if we set D 7 := max{C 2 , 1/C 1 }.□ 3.2.Definition of the heights.In the following, we use ∆ n to denote the Siegel disk of f n for all n ⩾ 0, where f 0 ∈ IS α ∪ {Q α } with α ∈ B N and f n is obtained by applying the near-parabolic renormalization operator.
Definition (The heights).Let M ⩾ 1.For n ⩾ 0, we define There are many choices of the height h n .One of the candidates is B(αn+1) 2π + M .In order to apply Lemma 2.11(a) directly, we choose h n above so that h n > 1/α n .Similar to (2.3) (see Figure 3), we define Recall that ψ n is defined in (2.38).For n ⩾ 0 and i ⩾ 1, we denote under Exp is a punctured rounded disk centered at the origin with radius n , there exists a small open neighborhood of z on which f n can be iterated infinitely many times.By Lemma 2.15(b), there exists a small open neighborhood of Ψ n (z) ∈ V n 0 on which f 0 can be also iterated infinitely many times.Since each z ∈ V 0 n satisfies this property and 0 ∈ ∂V n 0 , it follows that V n 0 ⋐ ∆ 0 .By a completely similar argument, we have V i n ⋐ ∆ n for any i > 0 and n > 0. □ Note that the forward orbit of V i n is compactly contained in ∆ n for any n ⩾ 0 and i ⩾ 0.Moreover, the backward orbit of V i n is also compactly contained in ∆ n if the preimage under f n is chosen in ∆ n .In the following, we always assume that M ⩾ M 1 unless otherwise stated.
3.3.The location of the neighborhoods.For n ⩾ 0, each V 0 n ∪ {0} is a closed topological triangle13 whose boundary consists of three analytic curves.We use The superscripts 'l', 'r' and 'b' denote 'left', 'right' and 'bottom', respectively.See Figure 5. Similar naming convention is adopted to V i n and their forward images for all n ⩾ 0 and i ⩾ 0. For example, For simplicity, we denote the segment The 'left' and the 'right' end points of I 0 n are denoted by ∂ l I 0 n and ∂ r I 0 n respectively so that f n (∂ l I 0 n ) = ∂ r I 0 n .Similar naming convention is adopted to I i n and their forward images for all n ⩾ 0 and i ⩾ 0. In particular, by Lemma 2.15(a) we have n and ∂ r S i n be the smooth edges of S i n containing ∂ l V i n and ∂ r V i n respectively.Let k n = k fn ⩾ 1 be the integer introduced in Proposition 2.3, D 3 > 0 be a constant introduced in Lemma 2.11 and D n = D fn be the set defined in (2.9).Lemma 3.4 (see Figure 6).There exists a constant M 2 ⩾ 1 such that if M ⩾ M 2 , then for all n ∈ N, we have Proof.The proof is mainly based on applying Koebe's distortion theorem and the definition of near-parabolic renormalization.
(a) By the definition of near-parabolic renormalization, we have αn+1) .
By Lemma 3.2, ∆ n+1 contains the disk D(0, ς n ), where αn+1) .Therefore, is a well-defined univalent map with |g ′ (0)| = 1.If M is large enough such that ι n is much smaller than ς n , then by Theorem 3.1 the distortion of the circle g(∂D(0, ι n )) relative to ∂D(0, ι n ) can be arbitrarily small.Part (a) is proved if we notice that Φ n (I 0 n ) is the closure of a connected component of Exp −1 • g(∂D(0, ι n ) \ {ι n }).(b) Still by the definition of near-parabolic renormalization, we have , where g is defined in (3.4).On the other hand, by (3.4) and Theorem 3.1(b), we assume that M is large such that ι n is small and g(D(0, ς n )) ⊃ D(0, e 2π ι n ).According to Theorem 3.1(c), we assume further that M is large such that g([0, ς n )) ∩ ∂D(0, r) is a singleton for any 0 < r ⩽ e 2π ι n .Therefore, 27 e −2π(hn−1) .This proves Part (b).(c) By the definition of near-parabolic renormalization, we have . By Theorem 3.1, the Euclidean length of the arc Exp(β ′ n ) with end points g([0, ς n )) ∩ ∂D(0, ι n ) and g(ι n ) can be arbitrarily small if M is large enough.This proves Part (c).□ Let D 3 > 0 be introduced in Lemma 2.11.In the following we always assume that M ⩾ max{M 2 , D 3 + 1 2π log 4D7 27 + 2} unless otherwise stated.Then

The sequence of the curves is convergent
In this section, we define a sequence of continuous curves (γ i n ) n∈N in the Fatou coordinate planes with i ∈ N. The image of each γ i n under Φ −1 n is a continuous closed curve contained in the Siegel disk ∆ n of f n .We shall prove that (γ n 0 ) n∈N convergents uniformly to the boundary of ∆ 0 .4.1.Definition of the curves and its parametrization.For each n ∈ N, note that a n+1 = ⌊ 1 αn ⌋.Recall that We denote n be the arc in Φ n (∂ l S 0 n ) connecting u n with u ′′ n .See Figure 6.We first give the definitions of two curves γ 0 n (t) and γ 1 n (t), where t ∈ [0, 1], and then define the curves (γ i n (t)) n∈N inductively.Definition of γ 0 n : The curve γ 0 n (t) : [0, 1] → C is defined piecewise as following: Lemma 4.1 (See Figure 6).The map γ 0 n (t) : [0, 1] → C has the following properties: (a) γ 0 n and γ 0 n + 1 are simple arcs in Let χ n,0 := χ fn,0 be the antiholomorphic map defined in (2.8).
Let D 3 > 0 be the constant introduced in Lemma 2.11.
□ By (3.5) and Lemma 4.2(d), for any t ∈ [0, 1] and ζ ∈ Exp −1 (∂∆ n+1 ), we have Define γ i n inductively: For all n ∈ N and 1 ⩽ ℓ ⩽ i with i ⩾ 1, we assume that the curves γ ℓ n (t) : [0, 1] → C and γ ℓ n (t) : [0, 1] → C are defined and satisfy (a ℓ ) γ ℓ n is defined as in (4.5); The proof of Lemma 4.3 is completely similar to that of Lemma 4.2.Moreover, one can define the thickened curve γ ℓ n of γ ℓ n with ℓ = i + 1 as in (4.5) similarly.By the definition of γ i n , we have Lemma 4.4.For each t 0 ∈ [0, 1], there exist two sequences (t n ) n∈N with t n ∈ [0, 1] and (j n ) n⩾1 with 0 ⩽ j n ⩽ a n , such that for all n ⩾ 1 and all i ∈ N, The curves are convergent.Our main goal in this subsection is to prove: Proposition 4.5.There exists a constant K > 0 such that for all n ∈ N, we have In particular, the sequence of the continuous curves (γ n 0 (t) : [0, 1] → C) n∈N converges uniformly as n → ∞.
In order to estimate the distance between γ i 0 (t) and γ i+1 0 (t) with t ∈ [0, 1], we will combine the uniform contraction with respect to the hyperbolic metrics and some quantitative estimates (with respect to the Euclidean metric) obtained in §2.4.For any hyperbolic domain X ⊂ C, we use ρ X (z)|dz| to denote the hyperbolic metric of X.The following lemma appears in [Che19, Lemma 5.5] in another form.For completeness we include a proof here.
Lemma 4.6.Let X, Y be two hyperbolic domains in C satisfying diam (Re (X)) ⩽ A ′ and B δ (X) ⊂ Y , where A ′ and δ are positive constants.Then there exists a number 0 < λ < 1 depending only on A ′ and δ such that for any z ∈ X, Proof.For any fixed z 0 ∈ X, we consider the holomorphic function The proof is finished if we set λ := (2A ′ + δ)/(2A ′ + 2δ).□ Let X be a set in C and z 0 ∈ X.We use Comp z0 X to denote the connected component of X containing z 0 .Let D n be the set defined in (2.9).For n ∈ N, we define , where h n is the height defined in (3.3).Note that each D ′ n is a hyperbolic domain.Let ρ n (z)|dz| be the hyperbolic metric of D ′ n .We use len(•) and len ρn (•) to denote the length of curves with respect to the Euclidean and the hyperbolic metric ρ n (z)|dz| respectively.Lemma 4.7.Let A ′ > 0 and δ > 0 be two constants.Then there exist A > 0 and 0 < ν < 1 depending only on A ′ and δ such that for any piecewise continuous curve , by the definition of near-parabolic renormalization (see also (2.8)), we have By Lemma 2.14, we have B δ0 (χ i,ji (D i )) ⊂ D i−1 for a constant δ 0 depending only on the class IS 0 .Without loss of generality, we assume that δ 0 < 1. Combining (4.7) and (4.8), we have ) can be decomposed as: → (D ′ i−1 , ρ i−1 ), where ρi (z)|dz| is the hyperbolic metric of χ i,ji (D ′ i ).According to Proposition 2.6, we have diam (Re χ i,ji (D ′ i )) ⩽ k 1 .By Lemma 4.6, the inclusion map ) is uniformly contracting with respect to the hyperbolic metrics (and the contracting factor depends only on k 1 and δ 0 ).Since χ i,ji : n , it follows that there exists a constant A ′′ > 0 depending only on A ′ and δ (not on n) such that len ρn (ϑ n ) ⩽ A ′′ .Define By the uniform contraction of χ i,ji for 1 ⩽ i ⩽ n with respect to the hyperbolic metrics, there exists a constant 0 < ν < 1 depending only on k 1 and δ 0 such that Lemma 4.8.There exists K 1 > 0 such that for any n ⩾ 1 and any continuous curve A direct calculation shows that We claim that there exists K ′ 1 > 0 which is independent of α n such that In fact, a direct calculation shows that where dr, and We assume that α n is small such that 2πα n D ′ 2 ⩽ 1/2 and 2πα n D ′ 2 log(2 where r − 2D ′ 2 log(2 + r) ⩾ 4 if r ⩾ D ′ 6 (see Proposition 2.13(b)).By (4.12), there exist C 1 , C ′ 1 > 0 which are independent of α n such that For J 2 , since the integral is convergent, it follows that there exists a constant C 2 > 0 which is independent of α n so that J 2 ⩽ C 2 .Similarly, there exists a constant C 3 > 1 which is independent of α n so that J 3 ⩽ C 3 .Hence (4.11) follows if we set 6 , +∞).Therefore, ϕ 1 (r) and ϕ 2 (r) are monotonously decreasing on [ 1 4αn , +∞) and [D ′ 6 , 1 4αn ] respectively.Denote (4.13) Then ϕ(r) is monotonously (may not strictly) decreasing on [D ′ 6 , +∞).By Lemma 4.1(d), we have By (4.10) and (4.11) we have where The proof is complete.□ Proof of Proposition 4.5.Note that γ n 0 (t) = γ n 0 ( a1−1 a1 t) for all t ∈ [0, 1] and all n ∈ N. In order to prove (4.6), it suffices to prove that there exist K > 0 and a sequence of non-negative numbers (y i ) i⩾0 such that for any n ∈ N, any 0 ⩽ i ⩽ n and any t 0 ∈ [0, 1], we have We divide the argument into several steps.
Step 4. The conclusion.Since B δ (η n 0 ) ⊂ D ′ 0 , the Euclidean metric and the hyperbolic metric ρ 0 of D ′ 0 are comparable in a small neighborhood of η n 0 .Hence there exists a constant C > 0 depending only on δ such that Therefore, for all n ⩾ 0 we have By (4.13), (4.14) and the similar estimates to (4.24) and (4.25) in the above inductive procedure, it follows that for any n ⩾ 0, there exists a sequence of nonnegative numbers {y Then Proposition 4.9.The limit . By Lemma 3.2, the inner radius of the Siegel disk of f n+1 is c n+1 e −B(αn+1) , where c n+1 ∈ [D −1 7 , D 7 ] and D 7 > 1 is a universal constant.According to the definition of near-parabolic renormalization f n+1 = Rf n , there exists a point For any t 0 ∈ [0, 1] and n ⩾ 1, we choose ) is a continuous closed curve for all n ⩾ 0. Since γ n 0 (t) converges uniformly to the limit γ ∞ 0 (t) on [0, 1] as n → ∞, it follows that Φ −1 0 (γ ∞ 0 ) is a continuous closed curve which separates ∆ 0 from each component of U 0 \ ∆ 0 , where U 0 is the domain of definition of f 0 .In particular, we have Proof of the the first part of the Main Theorem.Suppose f 0 ∈ IS α ∪ {Q α }, where α ∈ B N with N sufficiently large.By Proposition 4.9, the boundary of the Siegel disk ∂∆ 0 = Φ −1 0 (γ ∞ 0 ) of f 0 is connected and locally connected.On the other hand, the Siegel disk ∆ 0 is compactly contained in the domain of definition of f 0 by Proposition 2.16(b).By the definition of ∆ 0 , there exists a conformal map ϕ : D → ∆ 0 so that f 0 • ϕ(w) = ϕ(e 2πiα w).According to Carathéodory, the map ϕ can be extended continuously to ϕ : D → ∆ 0 .

A Jordan arc and a new class of irrationals
In this section, we first define a Jordan arc Γ connecting the origin with the critical value cv = −4/27 in the domain of definition of f ∈ IS α ∪ {Q α } with α ∈ HT N .In particular, this arc is contained in P f .Then we define a new class of irrational numbers based on the mapping relations between the different levels of the renormalization.(5.1) and a topological triangle Lemma 5.1.There exists ε ′ 4 ∈ (0, ε 4 ] such that for all f ∈ IS α with α ∈ (0, ε ′ 4 ], Q f \ {0} ⊂ D(0, 4 27 e 3π ) \ [0, 4 27 e 3π ). (5.2) We postpone the proof of Lemma 5.1 to Appendix A. The inclusion relation (5.2) is proved for the maps in IS 0 first and then a continuity argument is used.
Corollary 5.2.For each n ⩾ 1, there exists a unique anti-holomorphic inverse branch of the modified exponential map Exp: Proof.Since Exp takes the value −4/27 at each integer, it follows that Exp has an inverse branch Log defined on Q n \ {0} such that Log(−4/27) = 1 since Q n \ {0} is simply connected and avoids the origin.By Lemma 5.1, we have Re and a topological triangle for every n ⩾ 0: Definition (see Figure 7).Let K . By Corollary 5.2, K n+1 ⊂ K n for all n ⩾ 0, the critical value cv = −4/27 is contained in the interior of K n and 0 ∈ ∂K n .Define Γ := n⩾0 K n .
(5.4) 7: A sketch of the renormalization microscope between levels 0 and 1.The sets Γ, 0 ′ , Q ′ n , K n with n = 0, 1 and some special points are marked.
Proof.The general idea of the proof is to use the uniform contraction with respect to the hyperbolic metrics to prove that Γ ∪ {0} is locally connected and then prove that it must be a Jordan arc.Let us prove it in details.
Therefore, the following convergence is uniform for t ∈ [0, +∞): Note that 1 ∈ 0 and Log • Φ −1 n (1) = 1.By the uniformly contracting of Log By a similar argument as above, we have Note that Γ is the intersection of the nested sequence (K n ) n⩾0 , where K n is the bounded component of C \ (Γ n 0,+ ∪ Γ n 0,− ∪ {0}) for all n ⩾ 0. Therefore, Γ = Γ ∞ 0,+ = Γ ∞ 0,− and Γ ∪ {0} is a Jordan arc connecting −4/27 with 0. □ 5.2.Dynamical behavior of the points on the arcs.Let ϕ 0 := id.For each n ⩾ 1, we denote Let Γ be the Jordan arc defined in (5.4).By the proof of Lemma 5.3, ϕ n can be defined on Γ 0 := Γ since where n ⩾ 1.Note that the restriction of Exp • Φ n−1 on Γ n−1 is a homeomorphism.Hence each Γ n ∪ {0} is also a Jordan arc connecting − 4 27 with 0 in the dynamical plane of f n .For each n ⩾ 1, the map ϕ n : Γ 0 → Γ n can be extended homeomorphically to is an unbounded arc in 0 ′ with the initial point 1.
In the following, we assume that α = α 0 ∈ B N , where B N is the set of high type Brjuno numbers defined in (2.39).Let B(α n ) be the Brjuno sum defined in (3.2).
Definition.Let H N be a subset of B N defined as ) .
In the next section we show that H N is independent of the choice of f 0 ∈ IS α ∪ {Q α } by proving that H N coincides with the set of high type Herman numbers.

Optimality of Herman condition
Herman condition is not easy to verify in general.Yoccoz gave this condition an arithmetic characterization so that one can check easily whether an irrational number is of Herman type.In this section, we first recall Yoccoz's characterization and then prove that under the high type condition, an irrational number is of Herman type if and only if it belongs to the set H N defined in §5.3.6.1.Yoccoz's characterization on H.For α ∈ (0, 1) and x ∈ R, define The map r α is of class C 1 on R, satisfying r α (log 1 α ) = r ′ α (log 1 α ) = 1 α , x + 1 ⩽ r α (x) ⩽ e x for all x ∈ R, and r ′ α (x) ⩾ 1 for all x ⩾ 0.
For an irrational number α ∈ (0, 1), we use (α n ) n⩾0 to denote the sequence of irrationals defined as in (2.35).Let B(α) be the Brjuno sum of α (see (3.1)).A Brjuno number α is a Herman number (or belongs to Herman type) if every orientation-preserving analytic circle diffeomorphism of rotation number α is analytically conjugate to a rigid rotation.Let H be the set of all Herman numbers.Theorem 6.1 ([Yoc02, §2.5]).Herman condition has the following arithmetic characterization: 6.2.Two conditions are equivalent.In this subsection, we prove that the set of Herman numbers is equal to H N defined in §5.3 under the high type condition.Lemma 6.2 ([Yoc02, Lemma 4.9]).Let α be irrational and x ⩾ 0. Then α ̸ ∈ H if and only if there exist m and an infinite set I = I(m, x, α) ⊂ N such that, for all k ∈ I, we have r α m+k−1 • • • • • r αm (x) < log 1 α m+k .Let D 4 and D 5 > 1 be the constants introduced in Lemma 2.12.
Definition.For α ∈ (0, 1) and y ∈ R, we define Let γ α = γ α0 be the unbounded arc defined in (5.8) and s αn := Φ n • Exp : γ n−1 → γ n the map defined in (5.9).By Lemma 2.12 and the definition of s α , we have the following immediate result.where s αn is the map defined in (6.1).We claim that x m+k−1 ⩾ 2πy m+k−1 + C 0 for all k ⩾ 0. (6.4) Assume that (6.4) holds temporarily.Since γ α is an arc starting at the point 1 and finally going up to the infinity, there exists ζ ∈ γ αm−1 so that Im ζ = 1.For k ⩾ 1, we denote where each s αn is defined in (5.9).By Lemma 6.3, we have y m+k−1 ⩾ Im ζ m+k−1 for all k ⩾ 1.
Since α ∈ H N , by the definition of H N and Lemma 5.4, there exists an integer k 0 ⩾ 1 such that for all k ⩾ k 0 , one has On the other hand, since α ̸ ∈ H N , by (6.3) there exists k ∈ I with k ⩾ k 0 such that x m+k−1 < log 1 α m+k .This is a contradiction since by (6.4) we have x m+k−1 ⩾ 2πy m+k−1 + C 0 > log 1 α m+k .Hence it suffices to prove the claim (6.4).Obviously, (6.4) is true when k = 0 since C 0 ⩾ 2π.Suppose x m+k−1 ⩾ 2πy m+k−1 + C 0 for some k ⩾ 0. It suffices to obtain x m+k ⩾ 2πy m+k + C 0 .The arguments are divided into following three cases.
Proof.The proof is similar to that of Lemma 6.where s αn is the map defined in (6.7).We claim that if C 0 is large enough, then 2πy m+k−1 ⩾ x m+k−1 + C 0 for all k ⩾ 0. (6.11) Assume that (6.11) holds temporarily.By Lemma 6.5, we have y m+k−1 ⩽ Im ζ m+k−1 for all k ⩾ 1.By (6.9), there exists an integer k ∈ I ′ with k ⩾ k 0 such that On the other hand, by (6.10), we have x m+k−1 ⩾ B(α m+k ).However, by (6.11) we have x m+k−1 ≤ 2πy m+k−1 − C 0 < B(α m+k ), which is a contradiction.Hence it suffices to prove the claim (6.11).Obviously, (6.11) is true when k = 0. Suppose 2πy m+k−1 ⩾ x m+k−1 + C 0 for some k ⩾ 0. Then one can divide the arguments into three cases as in Lemma 6.4 to obtain 2πy m+k ⩾ x m+k + C 0 .We omit the details since the rest proof is completely the same.□ Remark.In fact, if α ∈ H N , then according to [Ghy84] and [Her85], the boundary of the Siegel disk of each f ∈ IS α ∪ {Q α } contains the unique critical value − 4 27 .This implies that α ∈ H N by Proposition 5.7.Therefore in this way we also obtain Proof of the second part of the Main Theorem.Let α ∈ HT N be an irrational number of sufficiently high type.By Lemmas 6.4 and 6.6, α ∈ H N if and only if α ∈ H N .By Proposition 5.7, α ∈ H N if and only if cv = f (cp f ) ∈ ∂∆ f , where ∆ f is the Siegel disk of f ∈ IS α ∪ {Q α } and cp f is the unique critical point of f .Therefore, α ∈ H N if and only if cp f ∈ ∂∆ f .□ Appendix A. Some calculations in Fatou coordinate planes In this appendix we give the proof of Lemma 5.1 based on some estimates in sequence of the curves is convergent 25 5.A Jordan arc and a new class of irrationals 35 6. Optimality of Herman condition 41 Appendix A. Some calculations in Fatou coordinate planes 45

Figure 1 :
Figure 1: The domains U (the gray part), U ′ (the white region bounded by the blue curves, see (2.5) for the definition) and their successive zooms near −1.The outer boundary of U ′ looks like a circle with radius about 35 and the rightmost point of U is about 32.2.The widths of these pictures are 72, 0.6 and 0.0075 respectively.It can be seen clearly from these pictures that U ∩ (−∞, −1] = ∅ and U ⋐ U ′ . 17

Figure 2 :
Figure 2: The perturbed Fatou coordinate Φ f and its domain of definition P f .The image of P f under Φ f has been colored accordingly by the same color on the right.The blue set on the left depicts the forward orbit of the critical point cp f .

Figure 3 :
Figure 3: Left: The sets C f , C ♯ f and some of their preimages.The blue set depicts the forward orbit of the critical point cp f .Right: The images of C f ∪ C ♯ f and S f under the perturbed Fatou coordinate Φ f and it shows how the near-parabolic renormalization map is induced.

5 Exp
This map commutes with the translation by one.Hence it projects by the modified exponential map

Figure 4 :
Figure 4: The inverse Φ −1 f of the perturbed Fatou coordinate can be extended holomorphically to D f (colored cyan).It can be seen that the image Φ −1 f ( D f ) wraps around 0. The holomorphic map Φ −1f has an anti-holomorphic lift χ f such that Exp • χ f = Φ −1 f (note that Exp is anti-holomorphic).Some special points are also marked.
By definition we have 4 27 e −2πy ≍ α/e 2παIm ζ .A direct calculation shows that y = α Im ζ + 1 2π log 1 α + O(1), where O(1) is a number whose absolute value is less than a universal constant.(b) We divide the arguments into two cases.Firstly we assume that ζ ∈ D f with Re ζ ∈ [0, 1/(2α)].By Proposition 2.10(b), we have 2) Lemma 2.11 illustrates how the renormalization microscopes χ f reshapes the geometry of the Siegel disk at deeper scales.Specifically, Part (a) is for the points deep in the Siegel disk while Part (b) is for the points close to the Siegel boundary.
Hence there exists a small open neighborhood D of C ♯ n in P n such that Exp • Φ n (D) is compactly contained in the Siegel disk ∆ n+1 .By Lemma 2.15(c), it follows that f n can be iterated infinitely many times in D and the orbit is compactly contained in the domain of definition of f n .Note that 0 is contained in D. Therefore, D is contained in the Siegel disk of f n and C

Figure 5 :
Figure 5: In the dynamical plane of f n , the sets ∂ l V 0 n , ∂ r V 0 n and I 0 n are colored cyan, purple and red respectively.The blue set depicts the (partial) forward orbit of the critical point cp fn .The sets V 0 n and C ♯ n = f •kn n (V 0 n ) are colored gray.
[IS08],au-Fatou[Mil06,  §10]and Inou-Shishikura[IS08]).For all f ∈ IS 0 ∪ {Q 0 }, there exist two simply connected domains P attr,f , P rep,f ⊂ U f and two univalent maps Φ attr,f :P attr,f → C, Φ rep,f : P rep,f → C such that(a) P attr,f and P rep,f are bounded by piecewise analytic curves and are compactly contained in U f , cp f ∈ ∂P attr,f and ∂P attr,f ∩ ∂P rep,f = {0}; (b) The image Φ attr,f (P attr,f ) is a right half plane and Φ rep,f (P rep,f ) then any compact set K ⊂ P attr,f0 is contained in P fn for n large enough and the sequence (Φ fn ) converges to Φ attr,f0 uniformly on K; Moreover, any compact set K ⊂ P rep,f0 is contained in P fn for n large enough and the sequence (Φ fn − 1 α fn ) converges to Φ rep,f0 uniformly on K.
(4.19) holds if we set y i := inf n∈N y Remark.If α is of bounded type, or if there exists a universal constant C > 0 such that B(α n+1 ) ⩾ C/α n for all n ∈ N, then the sequence (γ n 0 (t)) n∈N converges exponentially fast as n → ∞.
where 0 ⩽ j i ⩽ a i and 1⩽ i ⩽ n.Re ζ n ⩽ a n+1 + k 1 + 2. (4.27)By Proposition 2.16(b), each Siegel disk ∆ n is compactly contained in the domain of definition of f n .For each n ∈ N, Φ −1 n is defined in D n (see Lemma 2.5).