Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery

In this paper we consider the complexification of the Arnold standard family of circle maps given by $\widetilde F_{\alpha,\epsilon}(u)=ue^{i\alpha} e^{({\epsilon}/{2}) (u-{1}/{u})}$, with $\alpha=\alpha(\epsilon)$ chosen so that $\widetilde F_{\alpha(\epsilon),\epsilon}$ restricted to the unit circle has a prefixed rotation number $\theta$ belonging to the set of Brjuno numbers. In this case, it is known that $\widetilde F_{\alpha(\epsilon),\epsilon}$ is analytically linearizable if $\epsilon$ is small enough and so it has a Herman ring $\widetilde U_{\epsilon}$ around the unit circle. Using Yoccoz's estimates, one has that the size$\widetilde R_\epsilon$ of $\widetilde U_{\epsilon}$ (so that $\widetilde U_{\epsilon}$ is conformally equivalent to $\{u\in{\mathbb C}: 1/\widetilde R_\epsilon < |u| < \widetilde R_\epsilon\}$) goes to infinity as $\epsilon\to 0$, but one may ask for its asymptotic behavior. We prove that $\widetilde R_\epsilon=({2}/{\epsilon})(R_0+\mathcal{O}(\epsilon\log\epsilon))$, where R0 is the conformal radius of the Siegel disk of the complex semistandard map $G(z)=ze^{i\omega}e^z$, where $\omega= 2\pi\theta$. In the proof we use a very explicit quasiconformal surgery construction to relate $\widetilde F_{\alpha(\epsilon),\epsilon}$ and G, and hyperbolic geometry to obtain the quantitative result.


Introduction
The complex standard family of self maps of C * = C \ {0} is given by the two-parameter family F α,ε (u) = ue iα e (ε/2) (u−1/u) , where α ∈ [0, 2π) and ε ∈ [0, 1). These maps are holomorphic in C * and the points at 0 and infinity are essential singularities (see [Ba,Ko1,Mak,Ke,Ko2,F]). For small ε, these functions are perturbations of the rotation of angle α with respect to the origin. The interest in this family relies on the fact that it is the extension to the complex plane of the well-known Arnold family of circle maps (see [Ar, dMvS]). Indeed, the unit circle C 1 FIGURE 1. Rational Arnold tongues in the parameter space of the standard family up to denominator 5 (note that these are sets with interior). The curves correspond to the irrational tongues for γ = ( √ 5 − 1)/2 and θ = 5 √ 2−1.
is invariant under F α,ε , and using the homeomorphism e 2πix between T 1 = R/Z and C 1 , the map F α,ε|C 1 becomes the Arnold family: It is clear that if ε ∈ [0, 1), f α,ε is an orientation-preserving diffeomorphism of T 1 , and, thus, for each pair of parameters (α, ε) the rotation number of f α,ε is well defined (see §2.1 for the definition of the rotation number). The rotation number measures the asymptotic rate of rotation of points of the circle. For instance, a rigid rotation of T 1 of the form T θ (x) = x + θ has rotation number θ . All throughout this paper we fix an irrational rotation number θ and always choose the parameters α and ε such that the rotation number of f α,ε is θ . More precisely, we choose our rotation number θ among the Brjuno set of irrational numbers, which contains all Diophantine numbers (see [PM] for a precise definition of these sets). In the (α, ε)-parameter space, the set of parameters with a given rotation number θ is called the Arnold tongue T θ . If θ is rational, its Arnold tongue is a set with interior, while if the rotation number is an irrational number θ , then T θ corresponds to a curve connecting ε = 0 and ε = 1, which is in fact the graph of a function ε → α(ε) with α(0) = 2πθ (see Figure 1). If θ is a Brjuno number, the curve α(ε) is known to be analytic for ε small enough [Ri, FG].
Moreover, if the rotation number θ is a Brjuno number we have that for ε small enough, the map f α(ε),ε is analytically linearizable (see [Y2,PM,Ri]). That is, there is an analytic map η ε : T 1 → T 1 that conjugates f α(ε),ε to T θ , i.e. f α(ε),ε • η ε = η ε • T θ . (2) Equivalently, to say that F α(ε),ε restricted to C 1 is analytically linearizable means that there is an analytic map ϕ ε : C 1 → C 1 , such that where R ω (u) = e iω u and ω = 2πθ. Since the linearization ϕ ε is analytic, it can be extended to a neighborhood of the unit circle of the form A(1/r, r), where we define A(r 1 , r 2 ) = {u ∈ C : r 1 < |u| < r 2 } as the straight ring of radii r 1 and r 2 . We denote by A ε = A(1/ R ε , R ε ) the maximal ring for which ϕ ε can be analytically continued. Then, it is easy to check that being F α(ε),ε|C 1 analytically linearizable is equivalent to the existence of a Herman ring U ε for F α(ε),ε , which is given by U ε := ϕ ε ( A ε ). In U ε , every orbit under F α(ε),ε lies on an invariant closed curve which has rotation number θ . Since ϕ ε is unique (up to composition with rotations), the constant R ε is univocally defined and we call it the size of the Herman ring.
The main goal of this paper is to give an asymptotic estimate for the size R ε of the Herman ring U ε as ε → 0.
The sharpest results concerning the size of Herman rings for univalent maps on a given ring are due to Yoccoz (see Theorems 2.1 and 2.3), who gives an estimate that can be applied to any analytic map F that leaves the unit circle C 1 invariant and has rotation number θ (i.e. a lift on R of F |C 1 has rotation number θ ), and which depends only on θ and on the size of the domain where the map is univalent. In §3 we will see that this general result applied to the complex standard family leads to where K = exp(− (θ) − 2πC 0 ), is the Brjuno function [MMY], C 0 is a universal constant and σ (ε) The fact that this estimate holds for any analytic diffeomorphism having C 1 invariant with rotation number θ and univalent at least in A(ε/σ (ε), σ (ε)/ε), suggests that a better estimate can be found for the complex standard family. We shall return to this problem in a moment, but first let us consider what is known as the complex semistandard map of parameter e iω G(z) = ze iω e z .
Observe that z = 0 is a fixed point of G with derivative e iω . Since ω = 2πθ, and θ is a Brjuno number, it is known [Br1,Br2] that G has a Siegel disk around the origin, which we denote by U . This means that if we call D r the open disk of center 0 and radius r, there exists a unique maximal number R 0 > 0 and a unique conformal isomorphism The number R 0 is known as the conformal radius of the Siegel disk. Standard arguments show that R 0 is always finite. Lower bounds for R 0 (as a function of ω) could be obtained applying Yoccoz's results [Y1] to the semistandard map. We now return to the problem of estimating the size of the Herman ring U ε . Our main result is the following theorem. THEOREM A. Let θ be a Brjuno number and consider the standard map F α,ε that F α(ε),ε restricted to C 1 has rotation number θ .
Let R ε be the size of its Herman ring and let R 0 be the conformal radius of the Siegel disk of the semistandard map G(z) = ze iω e z , where ω = 2πθ. Then, Remark 1.1. We believe that this is the best estimate that can be obtained with our methods. However, some recent developments (work in progress) seem to indicate that a more optimal estimate could be R ε = (2/ε)(R 0 + O(ε 2 )) (with R ε analytic on ε). As a vague indication, this could follow from knowing that the complexification of the Arnold tongue T θ , with θ a Brjuno number, can be parametrized holomorphically by a complex parameter strongly related with the modulus of the ring.
An analogue of Theorem A, for Chirikov's standard and semistandard maps of R 2 , was proved in [SV] using KAM methods and complex matching, and therefore restricting the result to Diophantine rotation numbers. In the present paper, we prove Theorem A using quasiconformal surgery, inspired by a qualitative construction of Geyer in [G] which relates the standard and the semistandard maps. We modify this construction by introducing the dependence on the parameter ε and by making most of its ingredients completely explicit. These additions will give us the possibility of obtaining quantitative estimates from the geometric construction.
From all the partial results involved in the proof of Theorem A we choose the following to be remarked on here, because of its interest in itself and its possible use in other surgery constructions. Part (a) is known from a theorem of Teichmüller, and relates the hyperbolic distance between the Morrey-Bojarski-Ahlfors-Bers map (see Theorem 2.12) and the identity with the dilatation µ . From this result, in part (b) we obtain an estimate for how close this map is from the identity.
Throughout the paper, δ U indicates the hyperbolic distance inside U, where U is a hyperbolic set (see §2.3). The notation · denotes the infinity norm and D = D 1 . PROPOSITION B. Let µ be a Beltrami coefficient on C (see §2.2) and h : C → C be the unique quasiconformal solution of the Beltrami equation ∂h/∂z = µ(∂h/∂z) fixing 0 and 1 (see Theorem 2.12).
This paper is organized as follows. Section 2 contains basic introductions to some of the tools and preliminary results that will be used during the proofs of Theorem A and Proposition B. The expert reader can go directly to §3, where the problem is scaled and restated more precisely. Sections 4 and 5 contain the actual proofs of Theorem A and Proposition B, respectively.

Preliminaries
In this section, we state the basic results that we need to prove Theorem A and Proposition B. In §2.1 we review previous results about the linearization of analytic circle maps and their translation to maps of the complex plane having an invariant circle. Section 2.2 is devoted to quasiconformal mappings and measurable Riemann mapping theorem. Finally, in §2.3 we give some definitions and results in hyperbolic geometry.
2.1. Analytic linearization. Let f : T 1 → T 1 be an orientation-preserving homeomorphism of the circle T 1 = R/Z, and f 0 its lift to R with the normalization f 0 (0) ∈ [0, 1). To such a map one can assign a rotation number defined as where x is any point in R. It is well known (see e.g. [dMvS]) that f being a homeomorphism guarantees that this limit exists and is independent of the point x.
With this definition, ρ(f ) is a rational number if and only if f has a periodic orbit. We are interested in maps with an irrational rotation number.
If the rotation number of f is an irrational number θ and f ∈ C 2 (T 1 ), Denjoy's theorem (see [dMvS]) ensures that f is topologically conjugate to the rigid rotation of angle θ , T θ (x) = x+θ . That is, there exists a homeomorphism η : T 1 → T 1 such that η•T θ = f •η, making the following diagram commute: If we require η(0) = 0, then the conjugacy is unique. From now on we restrict ourselves to the case where f is an analytic diffeomorphism of T 1 , and therefore it can be extended to a complex annulus, of certain width > 0, around T 1 : Abusing notation, we again denote this extension by f . If the conjugacy η is also analytic, the map f is said to be analytically linearizable. Then again, η can be extended to a neighborhood of the circle, and it is easy to check (by the principle of analytic continuation) that its extension also conjugates f to T θ wherever η is defined. We are particularly interested in the case where F : U ⊂ C → C is an analytic map having the unit circle C 1 invariant, and f is the map on T 1 induced by F |C 1 . In this case, we say that F |C 1 is analytically linearizable if there exists an analytic diffeomorphism ϕ : where R ω (u) = e iω u and ω = 2πθ. If we ask ϕ(1) = 1, ϕ is univocally defined and the relation between η and ϕ is given by The image by ϕ of the maximal ring where ϕ can be analytically continued is called the Herman ring of F . If R is the outer radius of this ring (in the understanding that this ring is symmetric with respect to the unit circle, and then it is of the form A(1/R, R)), the width of the annulus of analyticity of f around T 1 is (1/2π) log R. The quantity (1/π) log R is called the modulus of the ring and we call R the size of the ring.
Arnold showed in [Ar] that if θ is a Diophantine number and f is close enough to the rigid rotation T θ , then f is analytically linearizable. This result was later improved by Rüssmann [Ru1,Ru2], Herman [Her1,Her2] and Yoccoz [Y2,PM]. The sharpest results are due to Yoccoz and we state them below. In the statement, : R \ Q → R + ∪ {∞} denotes the Brjuno function, a purely arithmetic Z-periodic function. Its most important property is that (α) is finite if and only if α is a Brjuno number (see [MMY] for details on ).
The local conjugacy theorem, due to Yoccoz, states that any analytic circle map with a Brjuno rotation number and which is univalent in a sufficiently large annulus (where 'large' is defined only in terms of the rotation number) is analytically linearizable. Moreover, it gives a lower bound for the linearization domain which, again, only depends on the initial domain of univalency and the rotation number.
THEOREM 2.1. (Local conjugacy theorem) Let θ be a Brjuno number and > 0 such that Let f : T 1 → T 1 be an analytic circle diffeomorphism, orientation preserving and with rotation number θ .
We assume that f is holomorphic and univalent in the annulus A (see (6)). Then, f is analytically linearizable and the linearization η : Remark 2.2. For the Arnold standard family (1) a sort of reciprocal is also true. Indeed, it was shown in [G] that if a member of the Arnold standard family is analytically linearizable, then its rotation number must be Brjuno.
If F is a holomorphic map leaving the unit circle invariant, then by applying Theorem 2.1 to the map f induced by F |C 1 , we can state an analogous result about the analytic linearization of F . THEOREM 2.3. Let θ be a Brjuno number and R > 1 such that R > e (θ)+2πC 0 . Let F : C 1 → C 1 be an analytic diffeomorphism with rotation number θ . We assume that F is holomorphic and univalent in the ring A(1/R, R). Then, F is analytically linearizable and the linearization ϕ :

Quasiconformal mappings and the Beltrami equation.
In this section we briefly recall the relevant definitions and results to be used in the quasiconformal surgery procedure, which is going to be one of the main tools to prove the results of this paper.
The standard reference for quasiconformal mappings is [Ah]. In this section, U, V ⊂ C are open sets.
Equivalently, a Beltrami coefficient of U gives an almost complex structure σ , which means a measurable field of ellipses in the tangent space of U, centered at 0 and defined up to multiplication by a non-zero real constant. The argument of the major axis of these infinitesimal ellipses, at the point z ∈ U, is π/2 + arg(µ(z))/2, and the ratio of minor and major axes equals (1 − |µ(z)|)/(1 + |µ(z)|).
is a k-Beltrami coefficient. In this case, we say that µ f is the complex dilatation or the Beltrami coefficient of f .
Remark 2.6. With the same definition, but skipping the hypothesis on f to be a homeomorphism, f is called a k-quasiregular map. It is easy to check that a k-quasiregular map is locally the composition g•h of a holomorphic map g and a k-quasiconformal map h.
Definition 2.7. Given a Beltrami coefficient µ of V and a quasiregular map h : U → V, we define the pull-back of µ by h as the Beltrami coefficient of U defined by: Remark 2.9. Pulling-back by holomorphic functions does not increase the maximal dilatation, k, of a k-Beltrami coefficient.
Remark 2.10. The standard complex structure corresponds to µ 0 ≡ 0, which is a field of circles. A quasiregular mapping f is holomorphic if and only if f * µ 0 = µ 0 .
Definition 2.11. Given a Beltrami coefficient µ, the partial differential equation is called the Beltrami equation. By the integration of µ we mean the construction of a quasiconformal map f solving this equation almost everywhere or, equivalently, such that µ f = µ almost everywhere.
The famous measurable Riemann mapping theorem by Morrey, Bojarski, Ahlfors and Bers states that every almost complex structure is integrable. As we are going to use this result for Beltrami coefficients with U = V = C, we give a statement adapted to this context.
Furthermore, if µ t is a family of Beltrami coefficients such that µ t (z) depends analytically on t, for any z ∈ C, then h t depends analytically on t.
Remark 2.13. The application of the measurable Riemann mapping theorem to complex dynamics is the following. Let f and µ be, respectively, a quasiregular mapping of C and a Beltrami coefficient of C, such that f * µ = µ. If we apply Theorem 2.12 to integrate µ and we construct a quasiconformal mapping h such that µ h = µ, then g = h • f • h −1 verifies g * µ 0 = µ 0 , and hence g is a holomorphic map of C. Moreover, f and g are quasiconformally conjugate, i.e. they have the same dynamics.
2.3. Hyperbolic geometry. In this paper, besides quasiconformal surgery we will use some results of hyperbolic geometry (see [Be2] for a survey). Briefly, quasiconformal surgery will be the key for the geometrical constructions we do and hyperbolic geometry will provide some of the quantitative estimates.
Definition 2.14. Given U ⊆ C a domain (open and connected set) and given a continuous function λ : U → [0, +∞), with at most isolated zeros, we define the conformal metric λ on U as the metric having λ(z)|dz| as a line element. More precisely, given a piecewise differentiable arc γ : [a, b] → U, the length of γ with respect to the metric λ is defined by Definition 2.15. Given a conformal metric λ on U and given two points z 1 , z 2 ∈ U, we define the distance d λ (z 1 , z 2 ) by In the case when this infimum is achieved by an arc γ * from z 1 to z 2 , this arc γ * is called a geodesic between z 1 and z 2 .
Any holomorphic map between two domains U and V can be used to transport a conformal metric on V to a conformal metric on U.
Definition 2.16. Given a holomorphic map f : U → V and a conformal metric λ on V, we define the pull-back of λ by f as the conformal metric on U given by With this definition f is a local isometry between (U, f * λ) and (V, λ), i.e. it preserves arc-lengths. If f is biholomorphic, then it is a global isometry.
The example that concerns us is the hyperbolic metric, which is a conformal metric defined on domains U that have the unit disk D := D 1 as a covering space, and which is preserved under conformal self-mappings of U. On D, the hyperbolic metric takes the following form.
Definition 2.17. The hyperbolic or Poincaré metric on D is the metric defined by The Poincaré metric λ D is the unique metric on D (up to multiplication by positive constants) invariant under conformal automorphisms of D.
We will need an explicit expression for the distance in D defined by the Poincaré metric.
PROPOSITION 2.18. Given w 1 , w 2 ∈ D, we have the following formula for the hyperbolic distance δ D in D: .
The pull-back process allows us to transport the Poincaré metric to any domain U that is conformally equivalent to D. Indeed, if ψ : U → D is a Riemann map, then the hyperbolic metric on U is given by An important example of this is the upper half plane, H, for which we can take ψ(z) = (z − i)/(z + i), obtaining the following result.
PROPOSITION 2.19. The hyperbolic metric in H is given by λ H (z) = 1/Im(z). In this case, the λ H -geodesics are vertical segments or arcs of circles orthogonal to the real axis.
The hyperbolic metric can also be transported to non-simply connected domains, by means of any universal covering map. Then, if U is a hyperbolic domain and ϕ is a universal covering for U, the hyperbolic metric λ U |dz| is given as above by (8).
Using that ϕ(z) = exp((z − 1)/(z + 1)) is a universal covering for the punctured disk D * = D \ {0}, we obtain the following properties for the hyperbolic metric in D * .
The hyperbolic distance δ D * satisfies where we have chosen appropriate determinations for log z 1 and log z 2 (with the arguments of z 1 and z 2 differing at most by π). Moreover, the geodesics in D * are obtained by mapping the geodesics of H by the covering ϕ : H → D * given by ϕ(z) = e iz .
The main reason why hyperbolic geometry is very useful in complex dynamics is the fact that all holomorphic maps are contractive, when we look at them under the hyperbolic metric. This is known as the Schwarz-Pick lemma which reads as follows.
THEOREM 2.23. (Schwarz-Pick lemma) If U and V are hyperbolic domains and f : In this paper, we need to compare (locally) different hyperbolic distances. The following result, known as Ahlfors' lemma, gives a comparison between hyperbolic metrics. Since we are unable to provide a standard reference, we include its proof, taken from [Pet]. The analogous comparison for hyperbolic distances (which is in fact what we really need in the proof of Proposition B) requires some work and it is therefore given in Proposition 5.1 (see §5.2). PROPOSITION 2.24. (Ahlfors' lemma) Let U ⊆ V ⊂ C be hyperbolic domains. Then for any point z ∈ U, Proof. The left-hand inequality is quite immediate if we consider the identity map Id : U → V. By the Schwarz-Pick lemma, λ V (z)/λ U (z) ≤ 1 and we are done. For the right-hand inequality, let ϕ : D → V be a universal covering of V such that ϕ(0) = z and let 0 ∈ U ⊂ D be such that ϕ : U → U is conformal (see Figure 2).
Since ϕ is a (local) isometry between the hyperbolic metrics, we may work with D, U and 0 instead of V, U and z. In particular, we have Let r = min{|z| : z ∈ ∂U }. If r = 1 then U = D and there is nothing to prove. Hence we suppose r < 1. We now apply the left-hand inequality to D r ⊆ U to obtain and hence, It remains to show that 1/r = coth(d/2) where d = δ V (z, ∂U). To see this, we use Proposition 2.18. In particular, we observe that δ D (0, ·) has radial symmetry, and thus we have Therefore e d = (1 + r)/(1 − r) and

The complex standard family and the semistandard map
The complex standard family with α ∈ [0, 2π) and ε ∈ [0, 1), is a family of holomorphic maps of C * onto itself, with essential singularities at 0 and infinity. The maps of the family are symmetric with respect to the unit circle, which is invariant under F α,ε . The singularities of the inverse map consist exclusively of the images of the two critical points of F α,ε (as F α,ε has no asymptotic values) which are located at

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Moreover, one can see that the standard map is univalent on a symmetric ring A(1/r ε , r ε ), where Note that r ε 2/ε as ε → 0, and so r ε tends to infinity as ε tends to zero. From now on, we fix a rotation number θ in the Brjuno set and consider the analytic curve α = α(ε) such that the rotation number of F α(ε),ε|C 1 is θ . Thus, for ε small enough (depending only on θ ), the standard map is under the hypothesis of the local conjugacy theorem of Yoccoz (Theorem 2.3) which assures the existence of a Herman ring of size where K = exp(− (θ) − 2πC 0 ), is the Brjuno function and C 0 is a universal constant.
To asymptotically estimate the value of R ε , we start by scaling the problem hoping that the scaled value of R ε has a finite limit as ε tends to zero. We perform the change of variables z = ε 2 u, and we obtain a new map This map shows the complex standard family as a perturbation of the semistandard map G(z) = ze iω e z , with ω = 2πθ, as long as z is far away from zero (recall that α(0) = ω). Note that the limit is a singular limit at z = 0, since an essential singularity is converted into a fixed point. The new scaled map F α(ε),ε leaves C ε/2 invariant and its critical points are now located at c ± (ε) = 1 2 (−1 ± 1 − ε 2 ) < 0, which approach 0 and −1 as ε tends to 0. We also change variables on the conjugation plane so that the map where ϕ ε is given in (3), is now the linearizing map of F α(ε),ε|C ε 2 . The map ϕ ε is defined from the ring A(ε 2 /4R ε , R ε ), with R ε := (ε/2) R ε , to the scaled Herman ring U ε := ε/2 · U ε (see Figure 3). We will actually compare the scaled standard family, F α(ε),ε (z), with the semistandard map, G(z). The qualitative and quantitative relationship between these maps will be explained by the surgery construction in the next section.
Remark 3.1. At this point, after scaling, Theorem A is equivalent to proving that where R 0 is the conformal radius of the Siegel disk U of the semistandard map G(z) = ze iω e z (see (5)). In particular, this result implies that R ε is a continuous function at ε = 0. Note that (12) means that Yoccoz's estimate (9) can be improved for the standard family by observing that R ε = (σ (ε)/ε)K(ε), with K(ε) = R 0 + O(ε log ε) and, hence,

Proof of Theorem A
The proof of Theorem A is based on an explicit (quantitative) version of the (qualitative) surgery construction [G] that relates a member of the (non-scaled) complex standard family F α(ε),ε , with the semistandard map G. First, in §4.1 we explain Geyer's construction, slightly modified and adapted to the scaled map F α(ε),ε . In §4.2 we re-formulate Theorem A in terms of the previous surgery construction. Section 4.3 gives an explicit version of Geyer's construction, which allows us to obtain the quantitative results. In §4.4 we obtain Theorem A as an easy consequence of Proposition B and the results of §4.3. Finally, §5 contains the proof of Proposition B.
4.1. Surgery construction. The idea of Geyer's construction to relate F α(ε),ε to G is basically to 'fill up the hole' of the Herman ring U ε in order to transform the Herman ring into a 'Siegel disk'. For our purposes, 'the hole' is the disk of radius ε/2 (denoted by D ε/2 ), given the fact that its boundary is simpler than the boundary of U ε and that it is invariant under the map. This can be accomplished by defining a new map H ε which consists of the old one F α(ε),ε everywhere outside D ε/2 , and a suitable quasiconformal map conjugate to a rotation of angle ω inside this disk. Due to the fact that the behaviour of the scaled standard map and the semistandard map at ∞ are the same, the map H ε thus obtained is then 'morally' equivalent (in the dynamical sense) to the semistandard map. However, the map H ε constructed in this way will be quasiregular, but not holomorphic. We shall make it holomorphic by means of the measurable Riemann mapping theorem (see Theorem 2.12), as explained in Remark 2.13. So, we construct a Beltrami coefficient µ ε , invariant by H ε , and by the integration of µ ε we will obtain a quasiconformal map h ε such that h ε • H ε • h −1 ε is holomorphic and has the 748 N. Fagella et al same dynamics as H ε (holomorphic smoothing). By choosing H ε appropriately, we prove that this new map is the semistandard map G.
We now proceed to make this construction precise. To define a rotation inside the small disk D ε/2 , we first choose a 'gluing' map ψ ε . Let ψ ε : D ε/2 → D ε/2 be any quasiconformal map that agrees with ϕ ε on the boundary (i.e. ψ ε|C ε/2 = ϕ ε ) and sends 0 to 0. Since ϕ ε is a real analytic map, the existence of ψ ε is guaranteed (see [Pom]). Then we define the new map H ε as Figure 4). By the choice of ψ ε , the map H ε is continuous and quasiregular. By construction, it has a fixed point at z = 0, and it is conjugate to a rotation of angle ω on the set (topological disk) D ε/2 ∪ U ε by means of the conjugacy maps ϕ ε and ψ ε , which match up continuously. Last, we note that H ε has only one critical point: the former critical point c − (ε) of F α(ε),ε (given in (10)) which was outside D ε/2 , since the symmetric one c + (ε) has been annihilated.
To start the second part of the surgery construction (holomorphic smoothing) we define the Beltrami coefficient µ ε on C as follows. First we define it on the surgery region by pulling back µ 0 = 0 to D ε/2 by means of ψ −1 ε . We extend this almost complex structure to every preimage of D ε/2 using H ε (or equivalently F α(ε),ε , as both maps coincide outside of D ε/2 ); and finally we set µ ε = 0 at the remaining points. That is, where H −n ε (D ε/2 ) should be understood as the set of points whose nth iterate falls (for the first time) in D ε/2 . Note that with this definition and using that H ε (U ε \ D ε/2 ) = U ε \ D ε/2 , we have that the points in U ε \ D ε/2 (and all their preimages) satisfy µ ε (z) = 0. By construction, we have that µ ε is measurable and invariant under the pull-back by H ε , for it is spread out by the dynamics.

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As explained in Remark 2.13, the composition map is not only quasiconformally conjugate to H ε but also holomorphic in C. Moreover, if µ ε < 1 3 , then we have that this map does not depend on the parameter ε as shown in the following proposition.
The proof of this proposition is similar to an analogous result for Geyer's construction in [G]. However, for the sake of completeness, we include here a proof adapted to our context. The key tool in the proof is an estimate on the growth of a quasiconformal map at ∞. This estimate is a consequence of the following basic property of quasiconformal maps.  Proof. We set h = φ −1 and g(z) = 1/h(1/z). It is easy to check that g is k-quasiconformal, with g(0) = 0 and g(∞) = ∞. Applying Theorem 4.4 we have that there exists a constant m > 0 such that We take z 1 = z and z 2 = 0, and we replace z = 1/φ(w) to obtain the desired bound: . However, this estimate holds provided that |φ(w)| ≥ 1.
Combining the first two properties of G ε , we have that with g ε entire and without zeros. Now, we can estimate the growth order of g ε . To this end, we use that if z ∈ C \ D ε/2 , then So, we have that if |z| 1, then From Lemma 4.5 we have that if |z| 1 there exists some constant C > 0 such that For the other values of z the map h ε is bounded. This is also true for h −1 ε , and both facts can be summarized by saying that there exists M > 0, which depends only on ε, such that Moreover, we may also ask that |h −1 ε (z)| ≥ 1 if |z| 1, obtaining where m 1 and m 2 may depend on ε (of course, the condition |z| 1 is not necessarily uniform on ε). As we are assuming µ ε < 1/3, we have that 1 ≤ K ε < 2, and so we deduce that g ε has growth order controlled by with 1 ≤ p < 2. The known properties of g ε (z) (entire function without zeros and with exponential growth of order 1 ≤ p < 2) imply that it is of the form (see [D]) with P ε (z) a polynomial of degree not greater than 1. Now, the proposition follows from the remaining properties. 2 Remark 4.6. Assuming that µ ε < 1 3 , we have just proved that G = h ε • H ε • h −1 ε , and so H ε and G are conjugated by h ε . Then, as the invariant curves are preserved by conjugation, we have that the rotation domain D ε/2 ∪ U ε of H ε is mapped by h ε to the Siegel disk U of G (see (5)). That is This concludes the surgery construction relating the (scaled) standard map F α(ε),ε and the semistandard map G. In the following section we see which quantities we need to estimate in order to obtain quantitative information from the surgery we just performed.
Remark 4.7. Let us observe that (h ε • ψ ε ) (0) does not depend on the particular quasiconformal map ψ ε used in the surgery construction (which is of course not unique). This allows us to compute this derivative by explicitly constructing a convenient ψ ε .
From the previous observations, Theorem A follows immediately from the next proposition.
PROPOSITION 4.8. With the previous notation, we have To prove Proposition 4.8, we study the quantity by means of the Cauchy integral formula. This will be done in §4.4. The estimates we use come from studying the quantities |ψ ε (z) − z| and |h ε (z) − z| or, equivalently, how far the maps ψ ε and h ε are from the identity map in a neighborhood of zero.
To obtain such an estimate for ψ ε we construct ψ ε explicitly in §4.3. The estimate for h ε is a direct application of Proposition B.

Explicit surgery construction.
The main purpose of this section is to explicitly construct the quasiconformal extension ψ ε used in the surgery construction of §4.1, and to give the explicit estimates that measure how far ψ ε is from the identity map.
Let us recall that we have a circle of radius ε/2 on which the real analytic (scaled) conjugacy ϕ ε is defined. Our goal is to find a quasi-conformal map ψ ε : D ε/2 → D ε/2 that extends ϕ ε .
We define the 'gluing map' ψ ε to be the most natural extension: the radial one. More explicitly, given z ∈ D ε/2 we define This map is clearly continuous, it agrees with ϕ ε on the boundary of D ε/2 and sends 0 to 0. We also observe that it leaves all circles in D ε/2 invariant. In Proposition 4.9 below, we prove that ψ ε is a quasiconformal mapping if ε is small enough (even more, it is C ∞ at all points except at z = 0). Moreover, this result shows that for small values of ε we have µ ε < 1 3 (this condition is needed to show that G ε = G, as stated in Proposition 4.3), and so all the results derived from the quasiconformal construction (see § §4.1 and 4.2) hold.
Our goal in the remainder of this section is to prove the following result.
PROPOSITION 4.9. There exists a constant C 1 > 0, independent of ε, such that for any ε small enough, ψ ε is a (C 1 ε)-quasiconformal mapping in D ε/2 , and it verifies Remark 4.10. One could also use an alternative, more dynamically meaningful, quasiconformal extension of ϕ ε given by ψ ε (z) = ϕ 2|z| (z). In this case, the map H ε would be explicitly given by H ε (z) = F α(2|z|),2|z| (z) = ze iα(2|z|) e z−z on D ε/2 . This extension also preserves circles in D ε/2 and, on each of these circles, it is the linearizing map of a scaled standard map. In this case, one can show that |ψ ε (z) − z| ≤ C 1 |z| 2 .
The proof of this proposition will be an easy consequence of the following lemma.
The proof of Lemma 4.11 is deferred to the end of the section.
Proof of Proposition 4.9. First of all, we stress that the estimates given by Lemma 4.11 are only valid if we evaluate ϕ ε (z) for |z| = ε/2. From the definition of ψ ε (z) (see (13)) this is precisely the case which we are interested in. Let us see that ψ ε is a quasiconformal mapping, and obtain a bound for its distortion.
Then, applying Lemma 4.11, we can bound So, if we assume ε to be small enough in order to have that 3 2 C 2 ε ≤ 1 2 , we can bound the distortion of ψ ε by Hence, ψ ε is a (C 1 ε)-quasiconformal mapping with C 1 = 3C 2 . In order to estimate how far ψ ε is from the identity map, we apply the first inequality of (15) (see Lemma 4.11), obtaining The rest of the section is dedicated to proving Lemma 4.11. For this purpose, we will need a diophantine-like bound for Brjuno numbers that is weaker than the Brjuno condition. More precisely, if θ is a Brjuno number, there exist constants c 1 , c 2 , depending only on θ , such that for any k ∈ Z \ {0}, the following inequality is satisfied (see [Br1,p. 140] this inequality leads to the following lemma. LEMMA 4.12. Let m(x) be a 1-periodic function with zero average, and θ ∈ R be a Brjuno number,hence verifying (16). We assume that m is analytic in the complex annulus A c 3 = {x ∈ C/Z : |Im(x)| < c 3 }, being c 3 > c 2 , and that B = sup x∈A c 3 |m(x)| < +∞. Consider the 1-periodic solution ξ(x) of the difference equation: Then, ξ is analytic in A c 3 −c 2 , and verifies Proof. First, we expand m in Fourier series: Then, using the bound of |m| in the annulus A c 3 , and that it has zero average, one has that its Fourier coefficients verify m 0 = 0 and |m k | ≤ Be −2πc 3 |k| if k = 0. On the other hand, writing ξ also in Fourier series, we can solve (17) for the coefficients of ξ , obtaining Moreover, condition ξ(0) = 0 gives ξ 0 = − k∈Z\{0} ξ k .

N. Fagella et al
Let us now take c 5 any constant, independent of ε, verifying 0 < c 5 < (1/2π) log(c 4 /2ε) − c 2 . We observe that c 5 can be taken arbitrarily large, provided that ε is sufficiently small and that, with this definition of c 5 , we always have the inequality ε ≤ (c 4 /2)e −2π(c 2 +c 5 ) .
Then, (iii) follows by combining this expression with (i) and (ii), the mean value theorem, |log |z 1 | − log |z 2 || = 1 ξ ||z 1 | − |z 2 ||, ξ ∈ |z 1 |, |z 2 | , and the bound The third part of Lemma 5.5 is very close to what it is stated in Lemma 5.2, but it is not exactly what we need: we want a bound for |z 1 − z 2 | depending just on z 1 and δ D * (z 1 , z 2 ). This requires the following a priori estimate.