Dilatations and exponents of quasisymmetric homeomorphisms

Given a quasisymmetric homeomorphism, we introduce the concept of quasisymmetric exponent and explore its relations to other conformal invariants. As a consequence, we establish a necessary and sufficient condition on the equivalence of the dilatation and the maximal dilatation of a quasisymmetric homeomorphism by using the quasisymmetric exponent. A classification on the elements of the universal Teichmüller space is obtained by using this necessary and sufficient condition.