Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle

Huaying Wei; Katsuhiko Matsuzaki

REVIEW 2 major objections 5 minor 19 references

Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every subset of the unit circle, the paper constructs a complex Teichmüller space interpolating between the universal Teichmüller space and its little subspace.

desk verdict A sound new family of Teichmüller spaces, but the complex manifold structure rests on two lemmas from an unpublished preprint that need to be included or made available. read the letter →

arxiv 1908.08798 v1 pith:ULF4TA56 submitted 2019-08-22 math.CV

Huaying Wei , Katsuhiko Matsuzaki This is my paper · ORCID

classification math.CV MSC 30F6030C6232G1537E1058D05

keywords TeichmüllerspaceuniversalpiecewisesymmetrichomeomorphismBersembeddingbarycentricextensionasymptoticquasiconformalmapping

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a family of Teichmüller spaces $T^X_\sharp$ parameterized by subsets $X$ of the unit circle, sitting between the universal Teichmüller space $T$ and its little subspace $T_0$. For a finite set $X$, $T^X_\sharp$ consists of quasisymmetric circle homeomorphisms that are symmetric on every complementary interval, hence 'piecewise symmetric.' The paper proves that each $T^X_\sharp$ is a complex Banach manifold: its Bers embedding is the bounded domain $\beta(T)\cap B^X_\sharp(D^*)$, where $B^X_\sharp(D^*)$ consists of Schwarzian derivatives vanishing at the boundary relative to $X$. It also proves that the quotient $T/\sim_X$ is a complex manifold modeled on the quotient Banach space $B^X_\sharp(D^*)\setminus B(D^*)$, providing an intermediate analog of the asymptotic Teichmüller space between $T_0$ and $T$.

What carries the argument

The machinery has three parts. First, $L^X_\sharp(D)$ is the closed subspace of $L^\infty(D)$ consisting of Beltrami coefficients that vanish at the boundary relative to $X$: for every $\varepsilon>0$ there is a compact set $K\subset D\cup X$ with $\|\mu|_{D\setminus K}\|_\infty<\varepsilon$. Its image under the Bers Schwarzian derivative map $\Phi$ is $B^X_\sharp(D^*)$, the analogous relative-vanishing subspace of holomorphic quadratic differentials on the exterior disk $D^*$. Second, the Douady–Earle barycentric extension supplies a conformally natural section of the Teichmüller projection, and the paper proves that this section carries $T^X_\sharp$ into $M^X_\sharp(D)$, the same behavior that is known for $T_0$. Third, a local holomorphic section of $\Phi$ is built from Ahlfors' quasiconformal reflection estimates, and right-translation automorphisms of the Beltrami space move that section to any point; the resulting split submersion structure is what makes the Bers embedding a homeomorphism onto a bounded domain and gives the complex atlas on $T^X_\sharp$.

What would settle it

Take two equivalent Beltrami coefficients $\mu,\nu\in M^X_\sharp(D)$ and compute the complex dilatation $\lambda=\mu \ast \nu^{-1}$ of $f_\mu\circ f_\nu^{-1}$; if $\lambda$ fails to lie in $M^{f_\nu(X)}_\sharp(D)$, the right-translation lemma underlying Theorem 6.1 is false, and the Banach manifold atlas on $T^X_\sharp$ collapses. A concrete finite-$X$ case is available because Proposition 3.1 decomposes $L^X_\sharp(D)$ into sectors, making the translation formula computable.

Watch

Extended reading notes

Core claim

The central claim is that prescribing where the boundary is allowed to be wild produces a genuine complex-analytic scale of Teichmüller spaces. For every subset $X$ of the circle, the space $T^X_\sharp = \mathrm{M\ddot{ob}}(S)\setminus QS^X_\sharp$, consisting of quasisymmetric boundary maps extendable by Beltrami coefficients that vanish at the boundary relative to $X$, is a complex Banach manifold under the Bers embedding, with image exactly $\beta(T)\cap B^X_\sharp(D^*)$. The quotient Bers embedding is well defined and is a homeomorphism onto its image in $B^X_\sharp(D^*)\setminus B(D^*)$, giving $T/\sim_X$ a complex Banach manifold structure. For finite $X$, the elements admit an intrinsic description as piecewise symmetric homeomorphisms, and as $X$ grows the spaces form a strictly increasing family that never exhausts $T$, even when the union of an increasing sequence of subsets is dense.

Load-bearing premise

The whole manifold structure rests on two unpublished technical lemmas about when composing quasiconformal maps preserves the condition that the distortion dies out near $X$; if either lemma has a counterexample, the complex atlas on $T^X_\sharp$ and the quotient structure built from it are unsupported.

Editorial extensions

If this is right

For a finite set $X$, the elements of $T^X_\sharp$ are exactly the normalized quasisymmetric homeomorphisms that are symmetric on every closed interval of $S\setminus X$, giving a purely boundary description without quasiconformal extensions.
The Bers embedding identifies $T^X_\sharp$ with the bounded domain $\beta(T)\cap B^X_\sharp(D^*)$, so $T^X_\sharp$ is a complex Banach manifold and the Teichmüller projection is a holomorphic split submersion.
The quotient $T/\sim_X$ carries a complex Banach manifold structure modeled on $B^X_\sharp(D^*)\setminus B(D^*)$, extending the asymptotic Teichmüller space construction to the intermediate scale.
Every $T^X_\sharp$ is contractible, matching the contractibility of the universal Teichmüller space.
For any strictly increasing sequence $X_1\subsetneq X_2\subsetneq\cdots$, the union $\bigcup_n T^{X_n}_\sharp$ is not closed in $T^X_\sharp$ for $X=\bigcup_n X_n$; in particular the universal Teichmüller space is not exhausted by piecewise symmetric spaces from a countable increasing family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

One extension the authors do not spell out is that the assignment $X\mapsto T^X_\sharp$ is a strictly monotone filtration of $T$ by boundary regularity, so the quotients $T^X_\sharp\setminus T$ form a family of intermediate asymptotic Teichmüller spaces that could index how much boundary regularity a marked conformal structure preserves.
Theorem 3.4 suggests that, for finite $X$, piecewise symmetry is a purely local condition on the circle; a natural testable extension is whether the same intrinsic characterization remains true for arbitrary $X$ without the finiteness assumption, using the relative vanishing condition as the definition of symmetry away from $X$.
The strictness results in Section 9 indicate that the topology of $X$ alone does not control $T^X_\sharp$; comparing $T^X_\sharp$ with $T^{\overline{X}}_\sharp$ for a dense set $X$ would clarify whether the construction depends only on the closure of $X$ or on finer data about how points accumulate.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper gives a genuinely new family of Teichmüller spaces T^X_♯, indexed by subsets X of the circle, sitting between the universal Teichmüller space T and its little subspace T0. The constructions are real and the proof style is careful.

The new material that earns its keep: Theorem 3.4 characterizes piecewise symmetric homeomorphisms intrinsically, purely by local symmetry on S\X, no quasiconformal extension mentioned. Theorem 5.1 packages the equivalences among vanishing Schwarzian differences, vanishing barycentric differences, and membership in the right T^Y_♯; this is the workhorse. Theorem 4.1, showing Φ sends M^X_♯ into B^X_♯, is a concrete adaptation of the Astala–Zinsmeister estimate, with the area argument written out. Section 7’s quotient Bers embedding for a non-group equivalence relation is a nice piece of work; Theorem 7.2 is not just a paraphrase of the asymptotic Teichmüller space case. Section 8 adds rigidity theorems, and Section 9 shows via Baire category that increasing unions of the T^{X_n}_♯ do not fill T even for dense X. All of that is solid.

The soft spot is exactly the one flagged in the stress-test. Theorem 6.1 is the theorem that makes T^X_♯ a complex Banach manifold, and its proof relies on two facts about right translations on M^X_♯, taken from the authors’ unpublished preprint [19]. The manuscript states them as [19, Lemma 6.1] and [19, Proposition 6.2] without proof. The transfer is not automatic because [19] works with a stronger decay condition (T^X_*), not the 'less restrictive' space T^X_♯. If those lemmas fail, the local holomorphic sections cannot be patched across fibers, and Corollary 6.2 and Theorem 7.2 lose their foundation. This is a genuine self-containedness gap, not a circularity: the construction of T^X_♯ does not depend on the theorems being proved. The rest of the argument does not appear to have any internal errors.

The paper is for people working in quasiconformal Teichmüller theory, especially on the little and asymptotic subspaces. It deserves a serious refereeing. My recommendation: accept for review; before final acceptance, require the authors to either include proofs of [19, Lemma 6.1] and [19, Proposition 6.2] in the paper or make the preprint available so the referee can check the transfer. This is fixable in revision, not a desk-reject situation.

Referee Report

2 major / 5 minor

Summary. The paper introduces a family of Teichmüller spaces T^X_♯, parameterized by subsets X of the unit circle, that interpolate between the universal Teichmüller space T and its little subspace T0. The space T^X_♯ is defined through Beltrami coefficients that vanish at the boundary relative to X, and, for finite X, is interpreted as the space of piecewise symmetric homeomorphisms. The main results are: an intrinsic characterization of piecewise symmetric homeomorphisms (Theorem 3.4); a Bers-image identity β(T^X_♯)=β(T)∩B^X_♯(D*) (Corollary 5.5); a claim that the Bers Schwarzian derivative map restricts to a holomorphic split submersion onto its image, making T^X_♯ a complex Banach manifold (Theorem 6.1, Corollary 6.2); a quotient Bers embedding for T/∼_X (Theorem 7.2); rigidity theorems for groups normalized by elements of QS^X_♯ (Theorems 8.2 and 8.3); and an exhaustion result for increasing sequences of subsets (Theorem 9.2, Proposition 9.3).

Significance. If the main results hold, this is a useful and natural family of intermediate Teichmüller spaces, with the quotient T/∼_X providing a new analog of the asymptotic Teichmüller space. The paper is carefully organized and follows standard techniques from Earle–Marković–Šarić and Gardiner–Sullivan. It gives a clean intrinsic characterization in Theorem 3.4 and a clean Bers-image identity in Corollary 5.5, and the rigidity section adds interesting applications. However, the central complex-manifold theorem depends on two lemmas quoted from the authors' unpublished preprint [19], and Section 9 contains a gap that appears to contradict Theorem 9.2. These issues are load-bearing and need to be addressed before the paper can be accepted.

major comments (2)

[Section 6, Theorem 6.1] The proof of Theorem 6.1, specifically the step moving a local holomorphic section from the barycentric representative μ=s([μ]) to an arbitrary equivalent point, relies on [19, Lemma 6.1] and [19, Proposition 6.2]. These two facts assert that right translations r_ν are biholomorphic automorphisms of M^X_♯ for ν∈M^X_♯ with [ν]=[0], and that any two equivalent elements of M^X_♯ are connected by such a translation. These facts are exactly what allows local sections to be patched across fibres, so without them the split-submersion property, the complex Banach manifold atlas on T^X_♯ in Corollary 6.2, and the quotient manifold structure in Theorem 7.2 are unsupported. Since [19] is an unpublished preprint treating the related but stronger-decay space T^X_*, the transfer to M^X_♯ is not automatic and should either be proved in this paper or the theorem should be made explicitly conditional on the acceptance of [19].
[Section 9, Proposition 9.3(2)] The proof of Proposition 9.3(2) shows only that for every ε>0 the truncated coefficient μ_K=μ·1_K satisfies ‖μ−μ_K‖<ε and μ_K∈M^{X_n}_♯ for some n. That places μ in the closure of ∪_n M^{X_n}_♯, not in the union ∪_n M^{X_n}_♯. The conclusion M^X_♯ ⊂ ∪_n M^{X_n}_♯, and hence T^X_♯ = ∪_n T^{X_n}_♯, does not follow. Moreover, the asserted equality is incompatible with Theorem 9.2, which applies to the same sequence X_n→X and concludes that ∪_n T^{X_n}_♯ is strictly contained in T^X_♯. The proposition should be corrected, presumably to a density or closure statement, and the surrounding discussion of exhaustion should be adjusted accordingly.

minor comments (5)

[Section 3, Proposition 3.2 and Remark 3.3] The notation 'open subset V of D with I⊂V' is ambiguous when D is the open unit disk, since a boundary interval I is not contained in D. Please specify that V is an open subset of the plane, or of the closed disk, containing the relevant boundary interval.
[Section 3, Proposition 3.1] The sentence 'The inclusion ⊃ is easy to see' is confusing; please state explicitly which inclusion is meant, for example L^X_♯(D) ⊃ ∑_i L^{ξ_i}_♯(D) or its reverse.
[Section 5, Corollary 5.4] The proof would be clearer if it explicitly described the homotopy: since s is a section of π, the contraction of M^X_♯ pulls back to a contraction of T^X_♯.
[References] References [17] and [19] are listed as 'to appear' and 'preprint'; please provide arXiv identifiers or publication status so that the cited lemmas can be checked by readers.
[Section 4, proof of Theorem 4.1] The estimate Area(γ_ζ(K)) ≲ 1−d(0,γ_ζ(K)) is used without comment; a brief justification would help, since K is an arbitrary compact set rather than a hyperbolic ball.

Circularity Check

1 steps flagged · score 4.0 of 10

Complex manifold structure on T^X_# rests on unpublished self-citation [19] for right-translation lemmas; otherwise the derivation is self-contained.

self citation load bearing [Section 6, paragraph before Theorem 6.1]
"we in particular see that rν is a biholomorphic automorphism of M X ♯ (D) for any ν ∈ M X ♯ (D) with [ν] = [0] (see [19, Lemma 6.1]). We also see that any equivalent Beltrami coefficients µ1, µ2 ∈ M X ♯ (D) are mapped to one another by a biholomorphic automorphism rν of M X ♯ (D) for some ν ∈ M X ♯ (D) with [ν] = [0] (see [19, Proposition 6.2])."

The proof of Theorem 6.1 constructs a local holomorphic section to Φ only at a barycentric representative and then must move it to an arbitrary equivalent Beltrami coefficient by the right translation rν. The two facts needed for this patching—that rν preserves M^X_# when [ν]=[0], and that any two equivalent coefficients in M^X_# differ by such an rν—are not proved in the paper. They are cited to [19], the authors' own unpublished preprint, which the introduction says concerns a different and more restrictive space T^X_*. Since Corollary 6.2 (the complex Banach manifold atlas on T^X_#) and Theorem 7.2 (the quotient complex structure) both depend on Theorem 6.1, the central manifold-structure claim is supported by an unverified self-citation rather than an independent derivation.

full rationale

The main construction T^X_# = π(M^X_#(D)) is a new definition and is not derived from its own target: the intrinsic characterization in Theorem 3.4 is proved using external results (Fehlmann; Earle–Markovic–Saric; Hu–Muzician), and the Bers-image identification β(T^X_#)=β(T)∩B^X_#(D*) is obtained from the equivalence theorem 5.1, whose proof is given in the paper. No fitted parameter is renamed as a prediction, and no target statement is used as an input. The one genuinely load-bearing self-citation occurs in Theorem 6.1: after constructing a local holomorphic section at a barycentric point, the proof needs to move the section to an arbitrary equivalent Beltrami coefficient by a right translation rν. The two facts needed—that rν preserves M^X_# when [ν]=[0] and that any two equivalent coefficients in M^X_# differ by such an rν—are not proved here; they are quoted from the authors' unpublished preprint [19], which the introduction says concerns a different, more restrictive space T^X_*. Corollary 6.2 (the complex Banach manifold atlas on T^X_#) and Theorem 7.2 (the quotient complex structure) therefore rest on this self-citation. Because the theorem's conclusion is not identical to its assumptions by construction, this is a load-bearing self-citation rather than definitional circularity; score 4.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper uses no fitted constants or empirical inputs. It relies on standard theorems of quasiconformal mapping and on several external results, including two lemmas from the authors' own unpublished preprint [19], which is the main unverified input.

assumptions (7)

standard math Measurable Riemann mapping theorem
Used throughout Section 2 to obtain quasiconformal homeomorphisms from Beltrami coefficients.
standard math Beurling-Ahlfors theorem: quasisymmetric iff quasiconformal extension
Cited in Section 2 as the foundation of the universal Teichmüller space.
domain assumption Fehlmann's characterization of local dilatation and symmetric homeomorphisms
Proposition 3.2 relies on [9, Staz 3.1] and [11, Proposition 3.1] for equivalence of local dilatation and symmetry.
domain assumption Hu-Muzician local asymptotic conformality of barycentric extension
Theorem 3.4 converse uses [13, Theorem 2] to obtain local asymptotic conformality of E(h) on intervals where h is symmetric.
domain assumption Earle-Markovic-Saric results on barycentric extension and Bers embedding
Theorem 5.1 and Section 6 rely on [8, Lemma 6.1] and the fact that the barycentric section maps T_0 into M_0.
standard math Ahlfors quasiconformal reflection estimates
Theorem 4.1 and Theorem 6.1 use the estimates from [1] on quasiconformal reflections and Schwarzian derivatives.
domain assumption Wei-Matsuzaki [19, Lemma 6.1 and Proposition 6.2] on automorphisms of M^X_#(D)
Theorem 6.1's split submersion proof assumes these unpublished lemmas from the authors' previous preprint; they assert that right translations by equivalent Beltrami coefficients are biholomorphic automorphisms of M^X_#(D).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle." pith.science (2026). https://pith.science/paper/ULF4TA56

@misc{pith2026190808798,
  author       = {Pith},
  title        = {Pith review of: Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ULF4TA56}},
  note         = {Machine review of arXiv:1908.08798}
}

abstract

We interpolate a new family of Teichm\"uller spaces $T_{\sharp}^X$ between the universal Teichm\"uller space $T$ and its little subspace $T_0$, which we call the Teichm\"uller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichm\"uller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichm\"uller space.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[19]

Wei and K

H. Wei and K. Matsuzaki, Teichm¨ uller spaces of generalized sym metric homeomorphisms, preprint. Department of Mathematics and Statistics, Jiangsu Normal U niversity 20 H. WEI AND K. MATSUZAKI Xuzhou 221116, PR China E-mail address : hywei@jsnu.edu.cn Department of Mathematics, School of Education, W aseda Uni versity Shinjuku, Tokyo 169-8050, Japan E-ma...

work page
[1]

Ahlfors, Quasiconformal reﬂections, Acta Math

L. Ahlfors, Quasiconformal reﬂections, Acta Math. 109 (1963 ), 291–301

work page 1963
[2]

Ahlfors, Lecture on Quasiconformal Mappings, Van Nostran d, 1966

L. Ahlfors, Lecture on Quasiconformal Mappings, Van Nostran d, 1966

work page 1966
[3]

Astala and M

K. Astala and M. Zinsmeister, Teichm¨ uller spaces and BMO, Math. Ann. 289 (1991), 613–625

work page 1991
[4]

Beurling and L

A. Beurling and L. V. Ahlfors, The boundary correspondence un der quasiconformal mappings, Acta Math. 96 (1956), 125–142

work page 1956
[5]

Cui, Integrably asymptotic aﬃne homeomorphisms of the circle and Teichm¨ uller spaces, Sci

G. Cui, Integrably asymptotic aﬃne homeomorphisms of the circle and Teichm¨ uller spaces, Sci. China Ser. A 43 (2000), 267–279

work page 2000
[6]

Douady and C

A. Douady and C. J. Earle, Conformal natural extension of hom eomorphisms of the circle, Acta Math. 157 (1986), 23–48

work page 1986
[7]

C. J. Earle and J. Eells, On the diﬀerential geometry of Teichm¨ ulle r spaces, J. Anal. Math. 19 (1967), 35–52

work page 1967

Show all 19 references

[8]

C. J. Earle, V. Markovic and D. Saric, Barycentric extension and the Bers embedding for asymptotic Teichm¨ uller space, Complex manifolds and hyperbolic geometry, Contemporary Math. vol. 311, pp. 87–105, Amer. Math. Soc., 2002

2002
[9]

Fehlmann, ¨Uber extremale quasikonforme Abbildungen, Comment

R. Fehlmann, ¨Uber extremale quasikonforme Abbildungen, Comment. Math. Helv. 56 (1981), 558– 580

1981
[10]

F. P. Gardiner and N. Lakic, Quasiconformal Teichm¨ uller theory, Mathematical Surveys and Mono- graphs, vol. 76, Amer. Math. Soc., 2000

2000
[11]

F. P. Gardiner and D. Sullivan, Symmetric structure on a closed c urve, Amer. J. Math. 114 (1992), 683–736

1992
[12]

T. W. Gamelin and R. E Greene, Introduction to Topology (2nd ed .), Dover, 1999

1999
[13]

Hu and O

J. Hu and O. Muzician, Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local, J. Anal. Math 117 (2012), 24 9–271

2012
[14]

Lehto, Univalent Functions and Teichm¨ uller Spaces, Springe r-Verlag, New York, 1986

O. Lehto, Univalent Functions and Teichm¨ uller Spaces, Springe r-Verlag, New York, 1986

1986
[15]

Matsuzaki, The universal Teichm¨ uller space and diﬀeomorph isms of the circle with H¨ older continuous derivatives, Handbook of Group Actions

K. Matsuzaki, The universal Teichm¨ uller space and diﬀeomorph isms of the circle with H¨ older continuous derivatives, Handbook of Group Actions. Vol. I, Adv. L ect. Math.(ALM), vol. 31, Int. Press, Somerville, MA, 2015, pp. 333–372

2015
[16]

Matsuzaki, Injectivity of the quotient Bers embedding of Te ichm¨ uller spaces, Ann

K. Matsuzaki, Injectivity of the quotient Bers embedding of Te ichm¨ uller spaces, Ann. Acad. Sci. Fenn. Math. 44 (2019), 657–679

2019
[17]

Matsuzaki, Rigidity of groups of circle diﬀeomorphisms and Teic hm¨ uller spaces, J

K. Matsuzaki, Rigidity of groups of circle diﬀeomorphisms and Teic hm¨ uller spaces, J. Anal. Math. (to appear), arXiv:1607.06316

arXiv
[18]

Nag, The Complex Analytic Theory of Teichm¨ uller Spaces, Wiley -Interscience, New York, 1988

S. Nag, The Complex Analytic Theory of Teichm¨ uller Spaces, Wiley -Interscience, New York, 1988

1988

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [19]

Wei and K

H. Wei and K. Matsuzaki, Teichm¨ uller spaces of generalized sym metric homeomorphisms, preprint. Department of Mathematics and Statistics, Jiangsu Normal U niversity 20 H. WEI AND K. MATSUZAKI Xuzhou 221116, PR China E-mail address : hywei@jsnu.edu.cn Department of Mathematics, School of Education, W aseda Uni versity Shinjuku, Tokyo 169-8050, Japan E-ma...

work page

[2] [1]

Ahlfors, Quasiconformal reﬂections, Acta Math

L. Ahlfors, Quasiconformal reﬂections, Acta Math. 109 (1963 ), 291–301

work page 1963

[3] [2]

Ahlfors, Lecture on Quasiconformal Mappings, Van Nostran d, 1966

L. Ahlfors, Lecture on Quasiconformal Mappings, Van Nostran d, 1966

work page 1966

[4] [3]

Astala and M

K. Astala and M. Zinsmeister, Teichm¨ uller spaces and BMO, Math. Ann. 289 (1991), 613–625

work page 1991

[5] [4]

Beurling and L

A. Beurling and L. V. Ahlfors, The boundary correspondence un der quasiconformal mappings, Acta Math. 96 (1956), 125–142

work page 1956

[6] [5]

Cui, Integrably asymptotic aﬃne homeomorphisms of the circle and Teichm¨ uller spaces, Sci

G. Cui, Integrably asymptotic aﬃne homeomorphisms of the circle and Teichm¨ uller spaces, Sci. China Ser. A 43 (2000), 267–279

work page 2000

[7] [6]

Douady and C

A. Douady and C. J. Earle, Conformal natural extension of hom eomorphisms of the circle, Acta Math. 157 (1986), 23–48

work page 1986

[8] [7]

C. J. Earle and J. Eells, On the diﬀerential geometry of Teichm¨ ulle r spaces, J. Anal. Math. 19 (1967), 35–52

work page 1967