Characterization of the Asymptotic Teichmüller Space of the Open Unit Disk through Shears

We give a parametrization of the asymptotic Teichmüller space AT (D) of the open unit disk through a space of equivalent classes of the shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the disk. Then using the parametrization, we define a 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). In addition, we revisit and give a complete proof to Šarić’s characterization of quasisymmetric homeomorphisms in terms of shear functions.