Continuity of the barycentric extension of circle diffeomorphisms of H\"older continuous derivatives

The barycentric extension due to Douady and Earle gives a conformally natural extension of a quasisymmetric automorphism of the circle to a quasiconformal automorphism of the unit disk. We consider such extensions for circle diffeomorphisms of H\"older continuous derivatives and show that this operation is continuous with respect to an appropriate topology for the space of the corresponding Beltrami coefficients.


Introduction
The barycentric extension due to Douady and Earle [5] gives a natural extension of a self-homeomorphism of the unit circle S to a self-homeomorphism of the unit disk D. It plays an important role applied to quasisymmetric homeomorphisms of S in the complex analytic theory of Teichmüller spaces. In this paper, we apply the barycentric extension to diffeomorphisms of S with Hölder continuous derivatives and obtain an analogous result for the Teichmüller space of such circle diffeomorphisms with the universal Teichmüller space.
The universal Teichmüller space T can be defined as the space QS * (S) of all normalized quasisymmetric homeomorphisms of S. In this setting, the Teichmüller projection q is regarded as the boundary extension map on the space QC * (D) of all normalized quasiconformal homeomorphisms of D. By the measurable Riemann mapping theorem, we can identify the latter space with the space of Beltrami coefficients Bel(D) = L ∞ (D) 1 , which is the open unit ball of measurable functions on D with the supremum norm. Then q : Bel(D) → T is continuous with respect to the topology on QS * (S) induced by the quasisymmetry constant. The barycentric extension yields a continuous section e : T → Bel(D) for q.
The Teichmüller space T α 0 of circle diffeomorphisms with α-Hölder continuous derivatives for α ∈ (0, 1) is similarly defined as a subspace of T ; the subgroup Diff 1+α * (S) ⊂ QS * (S) of all such diffeomorphisms with normalization can be defined to be T α 0 . The topology on this group is induced by the C 1+α -distance from the identity map. On the other hand, the corresponding subspace of Beltrami coefficients is Bel α 0 (D) ⊂ Bel(D), which consists of all µ ∈ Bel(D) with finite weighted supremum norm µ ∞,α = ess. sup ζ∈D 2 1 − |ζ| 2 α |µ(ζ)|.
Then we have proved in [12] that the restriction of the Teichmüller projection to Bel α 0 (D) gives a continuous map q : Bel α 0 (D) → T α 0 . In fact, the topology of T α 0 coincides with the quotient topology induced from Bel α 0 (D) by q. Moreover, a complex Banach manifold structure has been provided for T α 0 through the Bers embedding. See survey articles [10] for the introduction of the Teichmüller space T α 0 and [11] for applications of T α 0 to problems on circle diffeomorphism groups.
The main theorem of this paper asserts the continuity of the section e restricted to T α 0 . Theorem 1.1. The barycentric extension of circle diffeomorphisms with α-Hölder continuous derivatives gives a continuous section e : T α 0 = Diff 1+α * (S) → Bel α 0 (D) for the Teichmüller projection q.
As a well-known consequence from the existence of a continuous section, we understand a topological structure of this space. Note that T α 0 = Diff 1+α * (S) is also a topological group [12].
Corollary 1.2. The Teichmüller space T α 0 is contractible. In the next section, we will explain the above mentioned concepts and results in more detail.

Preliminaries
In this section, we summarize several results on the background of our arguments. This includes the definition and properties of the barycentric extension of quasisymmetric selfhomeomorphisms of the circle, fundamental results on the universal Teichmüller space and preliminaries on the space of circle diffeomorphisms with Hölder continuous derivatives. For the results mentioned in this section on quasiconformal and quasisymmetric homeomorphisms as well as Teichmüller spaces, we can consult the monograph by Lehto [9].
2.1. Quasiconformal and quasisymmetric homeomorphisms. We denote the group of all quasiconformal self-homeomorphisms of the unit disk D by QC(D) and the group of all quasisymmetric self-homeomorphism of the unit circle S by QS(S). Every f ∈ QC(D) extends continuously to a quasisymmetric homeomorphism of S. This boundary extension defines a homomorphism q : QC(D) → QS(S). Conversely, every ϕ ∈ QS(S) extends continuously to a quasiconformal homeomorphism of D, in other words, q is surjective. In fact, there are explicit ways of giving such quasiconformal extension which defines a section e : QS(S) → QC(D) with q • e = id| QS . The Beurling-Ahlfors extension [3] and the Douady-Earle extension [5] are well-known.

2.2.
The barycentric extension. The barycentric extension or the Douady-Earle extension e(ϕ) of an orientation-preserving self-homeomorphism ϕ ∈ Homeo(S) is given as follows. The average of ϕ taken at w ∈ D is defined by where the Möbius transformation sends w to the origin 0. The barycenter of ϕ is a point w 0 ∈ D such that ξ ϕ (w 0 ) = 0. This exists uniquely. The value of the barycentric extension e(ϕ) at the origin 0 is defined to be the barycenter w 0 ; we set e(ϕ)(0) = w 0 . For an arbitrary point z ∈ D, the barycentric extension e(ϕ) is defined by where γ ∈ Möb(D) is any Möbius transformation that maps 0 to z, say, γ = γ −1 z . This is well-defined since ξ ϕ•r (0) = ξ ϕ (0) for any rotation r, which is a Möbius transformation fixing 0.
An alternative definition was introduced by Lecko and Partyka [8]. For each w ∈ D, we consider the harmonic extension (the Poisson integral) of γ w • ϕ ∈ Homeo(S); Since P w is a self-homeomorphism of D by the Radó-Kneser-Choquet theorem, there exists a unique point z ∈ D such that P w (z) = 0. We define a map e * (ϕ) : D → D by e * (ϕ)(w) = z. Then e(ϕ) = e * (ϕ) −1 . Indeed, e(ϕ)(z) = w and e * (ϕ)(w) = z are equivalent to the conditions respectively. By substitutionζ = γ z (ζ), we see that these integrals are the same. The application of the barycentric extension to a quasisymmetric homeomorphism yields the following fundamental result.
Besides Douady and Earle [5], we can find an expository on the barycentric extension in Pommerenke [13,Section 5.5], which we consult occasionally hereafter.
On the other hand, QS * (S) can be regarded as the universal Teichmüller space T , which is equipped with the right uniform topology induced by the quasisymmetry constant M(ϕ) ≥ 1 for ϕ ∈ QS(S); a sequence ϕ n converges to ϕ in QS(S) if M(ϕ n • ϕ −1 ) → 1 (n → ∞). We note that there are several different ways of defining the quasisymmetry constant M, say, using the cross ratio, but they all induce the same topology.
Under the above identification, the restriction of q to QC * (D) = Bel(D) plays the role of the Teichmüller projection. A basic property of this projection is the following. is continuous and open.
The section for q given by the barycentric extension is also compatible with the topology. 2.5. Diffeomorphisms with Hölder continuous derivatives. An orientation-preserving diffeomorphism ϕ ∈ Diff(S) belongs to the class Diff 1+α (S) for α ∈ (0, 1) if its derivative is α-Hölder continuous. This means that the lift ϕ : R → R of ϕ given by for some constant c ≥ 0. We provide Diff 1+α (S) with the right uniform topology induced by C 1+α -distance p 1+α (ϕ) from id to ϕ ∈ Diff 1+α (S). Here Note that Diff 1+α (S) is a topological group with this topology [12].
Theorem. A quasisymmetric homeomorphism ϕ : S → S belongs to Diff 1+α (S) if and only if it has a quasiconformal extension f : D → D whose complex dilatation µ f belongs to Bel α 0 (D). "Only if" part was proved by Carleson [4] using the Beurling-Ahlfors extension of quasisymmetric functions on the real line. "If" part was investigated by Anderson and Hinkkanen [2] among others, and settled by Dyn'kin [6] and Anderson, Cantón and Fernández [1]. A different proof for an improved statement which is necessary to the arguments of Teichmüller spaces (Section 2.7) was given in [12].
2.7. The Teichmüller space for Diff 1+α (S). The previous theorem implies that the Teichmüller projection (boundary extension) gives a surjective map where the group Diff 1+α * (S) of the normalized elements can be defined to be the Teichmüller space T α 0 of circle diffeomorphisms with α-Hölder continuous derivatives. Moreover, taking the topology into account, we have proved the following.
Concerning the section given by the barycentric extension, we have also obtained that it has the right image.
Proposition ( [12]). The image of the barycentric extension of circle diffeomorphisms with α-Hölder continuous derivatives is contained in Bel α 0 (D).

An outline of the proof
This section is devoted to a sketch of the proof of our main theorem (Theorem 1.1). The arguments for the rigorous proof begins from the next section. Since the proof is rather technical and complicated, it will be helpful to mention its outline before.
We first give a set-up for the proof. Assuming the results in Section 2.7, we have only to prove the continuity of the barycentric extension e as in the following statement.
If e(ψ • ϕ 0 ) = e(ψ) • e(ϕ 0 ), the proof would be easy. But, the barycentric extension e is not a homomorphism; it only has the conformal naturality. We reduce the theorem to a simpler form by using the following facts: (1) Composition of a rotation does not change the derivatives of circle diffeomorphisms; (2) Post-composition of a Möbius transformation does not change the complex dilatations of quasiconformal homeomorphisms. Then we can normalize the situation so that ϕ 0 and ψ fix 1 and the derivative of ψ at 1 is 1, and we will estimate the complex dilatations on the real interval [0, 1) ⊂ D. Moreover, we have only to consider the convergence when |z| is sufficiently close to 1. Otherwise, 2/(1 − |z| 2 ) is bounded and the uniform convergence of complex dilatations follows from the convergence ψ → id by the arguments for the theorem in Section 2.4. Thus the above theorem is reduced to the claim below. The precise statement respecting the uniformity under conjugations by rotations will be given in Theorem 6.1 of Section 6.
The strategy for the proof is to use the conjugate by which maps the real interval [−1, 1] onto itself with the end points fixed and sends 0 to t. Then the conformal naturality of the barycentric extension implies that . From these equalities, the term in the above claim we are going to estimate becomes The advantage of this reduction is that we can explicitly represent µ e(ϕ) (0) for ϕ ∈ QS(S) by using the Fourier coefficients for ϕ (including the average of −ϕ 2 ) if e(ϕ)(0) = 0, that is, if Under this condition, we have This follows from [5, p.28]. See also [13, p.115]. However, there are also the following problems in these arguments: (1) How can we deal with the weight (2/(1 − t 2 )) α when t → 1.
(2) How can we estimate µ e(ϕ) (0) even if e(ϕ)(0) = 0; the barycenters of h −1 t • ϕ 0 • h t and h −1 t • ψ • ϕ 0 • h t are not necessarily zero. The solution to problem (1) is given by the precision of the following result due to Earle [7]: If ψ(1) = 1 and ψ ′ S (1) = 1 then h −1 t • ψ • h t converge to id uniformly on S as t → 1. This is because the conjugation by h t magnifies the mapping of ψ near 1, and since the linear approximation of ψ has slope ψ ′ S (1) = 1, it converges to the identity. Earle gave a more precise statement for it "with future applications in mind". We follow his arguments at the present by utilizing the α-Hölder constant of c α (ψ) (1). Integration of the definition of the α-Hölder constant (Proposition 4.1) yields This can make the above result by Earle to be a quantitative statement as follows. The proof will be given in Section 4.
Towards the solution to problem (2), we consider the barycenter e(ϕ t )(0) of the conju- Even if e(ϕ t )(0) = 0, we can estimate the Fourier coefficients for ϕ t uniformly if e(ϕ t )(0) is in a compact subset of D.
For the base point ϕ 0 ∈ Diff 1+α (S), the derivative (ϕ 0 ) ′ S (1) is not necessarily 1. In this case, the close-up of the behavior of ϕ 0 in a neighborhood of 1 by the conjugation of h t converges to the Möbius transformation h s satisfying (h s ) ′ S (1) = (ϕ 0 ) ′ S (1). More concretely, this is given in the following claim. The corresponding statement respecting the uniformity under normalization by rotation will be given in Lemma 4.2.
Fix t sufficiently close to 1. Then the claim says that ϕ t is uniformly close to h s . Under this condition, we can expect that the barycenter e(ϕ t )(0) should be close to e(h s )(0) = s, which is to be verified in Section 6. Hence, for some g 1 ∈ Möb(D) close to h s (written as g 1 h s ), we will have is close to h s . Hence, for some g 2 ( h s ) ∈ Möb(D), Now we represent the complex dilatations as Here a 1 , a −1 , b are the Fourier coefficients for g −1 1 • ϕ t and a ′ 1 , a ′ −1 , b ′ are the Fourier coefficients for g −1 2 • ψ t • ϕ t . By using the fact that g 1 g 2 , we can estimate |µ e(ψt•ϕt) (0) − µ e(ϕt) (0)| in terms of the approximation of h s by g 1 and g 2 . This will be carried out precisely in Section 6.

Convergence of conjugation of circle diffeomorphisms
In this section, we prepare certain results on the convergence of conjugation of circle diffeomorphisms by the canonical Möbis transformations, which is inspired by the paper of Earle [7]. These are necessary for the proof of our main theorem concerning the solution of the problems mentioned in the previous section.
In what follows, it is convenient to regard S being parametrized by arc length. For ζ 1 , ζ 2 ∈ S, the the length of the shorter circular arc connecting them is denoted by d S (ζ 1 , ζ 2 ). By the universal cover ζ = e ix : R → S, this is given by For ϕ 1 , ϕ 2 ∈ Homeo(S), we set Define ϕ : R → R to be a lift of ϕ ∈ Homeo(S) with exp(i ϕ(x)) = ϕ(e ix ). For ϕ ∈ Diff(S), its derivative along S at ζ = e ix is defined by ϕ ′ S (ζ) := ϕ ′ (x). The α-Hölder constant of ϕ at η = e iy ∈ S is given by First, we prepare an elementary fact on the integration of the α-Hölder continuity condition at 1 ∈ S.
For t ∈ (−1, 1), we utilize a particular Möbius transformation of D given by which maps the real interval [−1, 1] onto itself with the end points fixed and sends 0 to t. The following lemma, mentioned in Section 3, is an application of the arguments in Earle [7, Theorem 2] to an orientation-preserving self-homeomorphism ψ ∈ Homeo(S) approximating the identity with a prescribed order at the fixed point 1 ∈ S. The conjugate of ψ by h t expands the local behavior of ψ near 1 to the global S.
In the later application, we consider the situation where the constant c in Proposition 4.1, which will be taken as the α-Hölder constant c α (ψ)(1) of ψ ′ S at 1 ∈ S, can be arbitrarily small. Then we can choose the constant C in Lemma 3.2 as and apply the consequence of this lemma. We denote the rotation sending 1 to η ∈ S by r η ∈ Möb(S). The composition of rotations does not change the derivative at any point η ∈ S of a diffeomorphism ϕ 0 ∈ Diff 1 (S). Hence we may assume that it fixes 1. The previous lemma dealt with the case of its derivative at 1 is 1. The following lemma treats the general case and asserts the convergence of the conjugate by h t to an appropriate Möbius transformation.
Proof. Set ω = h t (ζ). Then We will estimate the difference between ϕ η 0 and h sη near 1. Note that ϕ η 0 (1) = h sη (1) = 1 and (ϕ η Claim. For anyε > 0, there existsδ > 0 independent of η such that if |h sη (ω) − 1| ≤δ then We consider the same estimate for the lift h sη of h sη . Since s η is uniformly bounded away from −1 and 1 (as ℓ η is uniformly bounded away from 0 and ∞) independent of η, we also have some constant On the other hand, since Therefore we obtain that Here, h sη (x) → 0 implies x → 0 and then the coefficient of | h sη (x)| in the last term tends to 0. Transforming this inequality for ϕ η 0 (ω) and h sη (ω), we can verify the required claim.

Average of circle homeomorphisms
The barycentric extension is defined by considering the average of a circle homeomorphism. In this section, we will show necessary properties of the average and the vector field given by the average function.
Recall that the Möbius transformation γ w ∈ Möb(D) is defined by First, we list up properties of γ w which will be used later. They are verified easily.
Proposition 5.1. The Möbius transformation γ w ∈ Möb(D) for each w ∈ D satisfies the following: (1) |γ w (z) − z| ≤ 2|w| 1 − |w| for every z ∈ D; (2) |γ ′ w (ζ)| = 1 − |w| 2 |ζ − w| 2 is the Poisson kernel, which satisfies For ϕ ∈ Homeo(S), we define its average taken at w ∈ D as Then ξ ϕ is a complex-valued differentiable function on D, which can be regarded as a vector field on D. If ϕ ∈ Homeo(S) is close to id, then the vector field ξ ϕ is close to ξ id as the following claim shows.
Proof. The definition of ξ implies that Then this is estimated from above by where the inner path integral is along the circular arc from ζ to ϕ(ζ). Since d S (ϕ(ζ), ζ) < ε, this integral is strictly bounded by ζ+ε ζ−ε |γ ′ w (η)| |dη|. Hence we have Here, the last equality is due to the fact that |γ ′ w (η)| is the Poisson kernel by Proposition 5.1 (2).
Proof. The barycenter w of ϕ satisfies ξ ϕ (w) = 0 by definition. Then the result follows from Proposition 5.2 and the remark after that.
We generalize the above proposition to an assertion on the difference between any two average functions and moreover on the difference between their derivatives. Proposition 5.4. For any ϕ, ψ ∈ Homeo(S), the following inequalities are satisfied for every w ∈ D: Estimating the absolute value of the denominator from below by (1 − |w|) 2 , we have the assertion.
(2) The ∂-derivative of ξ ϕ is and the same is true for ξ ψ . Then .
By the same estimate for the denominator as before, we have the assertion.
By the same estimate for the denominator as before, we have the assertion.
Next, we will see that if ϕ ∈ Homeo(S) is close to id and normalized so that its barycenter is at the origin 0 ∈ D, then |ξ ϕ (w)| can be estimated from below by |ξ id (w)| = |w| near the origin.

The proof of the main theorem
This section is entirely devoted to the proof of the main theorem in the form of Theorem 3.1. Actually, we first show that it can be reduced to Theorem 6.1 below. Then we aim to prove this theorem by dividing the arguments into several claims.
This relation can be alternatively written as .
We may assume that C η are uniformly bounded by some fixed positive constant, say, one. Then we can find a constant t 0 with 1 − δ 0 ≤ t 0 < 1 depending only on R, and hence only on ϕ 0 , such that for every η ∈ S and every t ∈ [t 0 , 1). Taking the supremum over η ∈ S and t ∈ [t 0 , 1), we have for some constant A > 0. This completes the proof of Theorem 6.1.