A generalized Stoilow decomposition for pairs of mappings of integrable dilatation

We prove a rigidity result for pairs of mappings of integrable dilatation whose gradients pointwise deform the unit ball to similar ellipses. Our result implies as corollaries a version of the generalized Stoilow decomposition provided by Theorem 5.5.1 of a recent monograph of Astala-Iwaniec-Martin and the two dimensional rigidity result of our previous paper for mappings whose symmetric part of gradient agrees. Specifically let $u,v\in W^{1,2}(\mathrm{\Omega },{\mathbb{R}}^2)$ where $\det (Du)>0$, $\det (Dv)>0$ a.e. and $u$ is a mapping of integrable dilatation. Suppose for a.e. $z\in \mathrm{\Omega }$ we have $Du(z{)}^{T}Du(z)=\lambda Dv(z{)}^{T}Dv(z)$ for some $\lambda >0$. Then there exists a meromorphic function $\psi$ and a homeomorphism $w\in W^{1,1}(\mathrm{\Omega }:{\mathbb{R}}^2)$ such that $Du(z)=\mathcal{P}(\psi (w(z)))Dv(z)$ where $\mathcal{P}(a+ib)=(\begin{array}{cc}a& -b\\ b& a\end{array})$. We show by example that this result is sharp in the sense that there can be no continuous relation between the gradients of $Du$ and $Dv$ on a dense open connected subset of $\mathrm{\Omega }$ unless one of the mappings is of integrable dilatation.