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We give a parametrization to the asymptotic Teichmuller space of the openunit disk through equivalent classes of shear functions induced byquasisymmetric homeomorphisms on the Farey tesselation of the unit disk. Thenusing the parametrization, we define a new metric on the asymptotic Teichmullerspace. Two other related metrics are also introduced on the asymptoticTeichmuller space by using cross-ratio distortions or quadrilateral dilatationsunder the boundary maps on degenerating sequences of quadruples orquadrilaterals. We show that the topologies induced by the three metrics areequivalent to the one induced by the Teichmuller metric on the asymptoticTeichmuller space. Before proving our main results, we revisit and rectify amistake in the proof in the reference [21] on the characterization ofquasisymmetric homeomorphisms in terms of shear functions.

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