Stoilow Factorization of the Heisenberg Group

In this article we study the properties of quasiconformal mappings on the Heisenberg group $\Bbb H^1$ and consider the definition of quasiconformal mappings in terms of the Beltrami equation. In particular, we obtain an explicit expression for the Beltrami coefficient for the composition of two quasiconformal mappings and we prove an analogue of the Stoilow factorization theorem on the plane. Namely, if the Beltrami coefficients of two quasiconformal mappings are equal almost everywhere, then there exists a conformal mapping such that by acting on one of the given quasiconformal mappings from the left, we obtain another given mapping. As an application of these results on the Heisenberg group $\Bbb H^1$ we compute the~Beltrami coefficients of some quasiconformal mappings and we prove a theorem on the images of quasi--Brownian motions. In specific examples we demonstrate the invariance of the~Beltrami coefficient under the action of the~composition of a conformal function on the corresponding left mapping. Using the Stoilow factorization on the~Heisenberg group, we show that if two quasi--Brownian motions have the corresponding Beltrami coefficients equal almost everywhere, then their trajectories are equivalent only if the conformal map in the Stoilov factorization is a map obtained from a composition of translations, rotations and dilations.