Abstract
A quasisymmetric homeomorphism of the unit circle s is called integrably asymptotic affine if it admits a quasiconfomal extension into the unit disk so that its complex dilatation is square integrable in the poincare medric on the unit disk, let QS * ( S 1 )be the space of such maps. here we give some characterizations and properties of maps in QS * ( S 1 ) .we also show that QS * ( S 1 )/mob ( S 1 ) is the completiono f diff( S 1 ) /mob(S1)in the weil-petersson metric.
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