Certain Integrability of Quasisymmetric Automorphisms of the Circle
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Certain Integrability of Quasisymmetric Automorphisms of the Circle Katsuhiko Matsuzaki
Received: 30 September 2013 / Accepted: 4 March 2014 / Published online: 24 June 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract Using the correspondence between the quasisymmetric quotient and the variation of the cross-ratio for a quasisymmetric automorphism g of the unit circle, we establish a certain integrability of the complex dilatation of a quasiconformal extension of g to the unit disk if the Liouville cocycle for g is integrable. Moreover, under this assumption, we verify regularity properties of g such as being bi-Lipschitz and symmetric. Keywords Quasisymmetric quotient · Cross-ratio · Quasiconformal map · Complex dilatation · Liouville cocycle · Asymptotically conformal Mathematics Subject Classification (2010) 37E30
Primary 30C62; Secondary 30F60 ·
1 Introduction A quasisymmetric automorphism g : S → S of the unit circle S = {ζ ∈ C | |ζ | = 1} plays a central role in the quasiconformal theory of Teichmüller spaces. For an orientation-preserving self-homeomorphism g of S, the concept of quasisymmetry can
In memory of Professor Gehring. Communicated by Bruce Palka. This work was supported by JSPS KAKENHI 24654035. K. Matsuzaki (B) Department of Mathematics, School of Education, Waseda University, Shinjuku, Tokyo 169-8050, Japan e-mail: [email protected]
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be defined in several ways, but boundedness of the variation of the following quantities under g is being used in some usual definitions, which are known to be equivalent: (1) the ratio of two intervals given by any three positively ordered points on S; (2) the cross-ratio of four positively ordered points on S. In both definitions, the variation is taken over all normalized points on S, i.e., three consecutive points with equal intervals in the first case and four points of even cross-ratio in the second case. To describe the above definition precisely, it is convenient to use a lift of an orientation-preserving self-homeomorphism g of S to a self-homeomorphism g of the real line R by the universal covering R → S with the correspondence x → ei x . In general, for an increasing homeomorphic function h : R → R, the quasisymmetric quotient is defined by m h (x, t) =
h(x + t) − h(x) h(x) − h(x − t)
for every x ∈ R and every t > 0. If m h (x, t) is uniformly bounded from above and away from zero, we say that h is quasisymmetric. More precisely, h is Mquasisymmetric if there exists a constant M ≥ 1 such that 1/M ≤ m h (x, t) ≤ M holds for every x ∈ R and for every t > 0. The quasisymmetry of g is defined by that of its lift g . Moreover, a quasisymmetric automorphism g of S or its lift g : R → R is called symmetric if m g (x, t) → 0 as t → 0 uniformly, i.e., independently of x ∈ R. On the other hand, for positively ordered distinct points ζ1 , ζ2 , ζ3 , ζ4 ∈ S, the cross-ratio is defined by [ζ1 , ζ2 , ζ3 , ζ4 ] =
(ζ1 − ζ3 )(ζ2 − ζ4 ) ∈ (1, ∞). (ζ1 − ζ4 )(ζ2 − ζ3 )
An orientation-preserving self-homeomorphism g of S is quasis
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