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

We give a parametrization to the asymptotic Teichmuller space of the open unit disk through equivalent classes of shear functions induced by quasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Then using the parametrization, we define a new metric on the asymptotic Teichmuller space. Two other related metrics are also introduced on the asymptotic Teichmuller space 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 Teichmuller metric on the asymptotic Teichmuller space. Before proving our main results, we revisit and rectify a mistake in the proof in the reference [21] on the characterization of quasisymmetric homeomorphisms in terms of shear functions.