We reinterpret our results in [Isr. J. Math. 159, 277–288 (2007; Zbl 1119.30024)] by using quasiconformal motions of the sphere, in the sense of Sullivan and Thurston. We show that a normalized holomorphic motion of a closed set E in the Riemann sphere extends to a normalized quasiconformal motion of the sphere if and only if the holomorphic motion is induced by a holomorphic map into the Teichmüller space of E, denoted by T(E). We also study a “universal” quasiconformal motion of the sphere and some other properties of continuous and quasiconformal motions.
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