Characterization of the asymptotic Teichmüller space of the open unit disk through shears

Abstract

We give a parametrization to the asymptotic Teichmüller space AT (D) of the open unit disk D through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of D. Then using the parametrization, we define a new metric on AT (D). Two other related metrics are also introduced on AT (D) by using cross-ratio distortions or quadrilateral dilatations under the boundary maps on degenerating sequences of quadruples or quadrilaterals. We show that the topologies induced by the three metrics are equivalent to the one induced by the Teichmüller metric on AT (D). Before proving our main results, we revisit and rectify a mistake in the proof in [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.2000 Mathematics Subject Classification. 30C75, 30F60.

Cross-ratio distortion and Douady–Earle extension: III. How to control the dilatation near the origin

“…The collection {f n } ∞ n=1 of the homeomorphisms f n in the proof of Proposition 1 provides a counter-example to Lemma 2.2 in [10]. By developing a suitable modification of that lemma, it is shown in [4] that the main results of [10] continue to hold.…”

“…In this paper, we need to consider quadruples Q with cr(Q) = 1. For each constant L ≥ 1, we define a cross-ratio distortion norm of f to be (4) ||f…”