Rigidity of pairs of quasiregular mappings whose symmetric part of gradient are close

For $A\in M^{2\times 2}$ let $S(A)=\sqrt{A^{T}A}$, i.e. the symmetric part of the polar decomposition of $A$. We consider the relation between two quasiregular mappings whose symmetric part of gradient are close. Our main result is the following. Suppose $v,u\in W^{1,2}(B_1(0):{\mathbb{R}}^2)$ are $Q$-quasiregular mappings with ${\int }_{B_1(0)}\det (Du{)}^{-p}dz\le C_p$ for some $p\in (0,1)$ and ${\int }_{B_1(0)}\mathrm{\mid }Du{\mathrm{\mid }}^{2}dz\le 1$. There exists constant $M>1$ such that if $${\int }_{B_1(0)}\mathrm{\mid }S(Du)-S(Dv){\mathrm{\mid }}^{2}dz=ϵ$$ then $${\int }_{B_{\frac{1}{2}}(0)}\mathrm{\mid }Dv-RDu\mathrm{\mid }dz\le c{C}_p^{\frac{1}{p}}{ϵ}^{\frac{p^3}{M{Q}^5\log (10C_{p}Q)}}\text{ for some }R\in SO(2)\mathrm{.}$$ Taking $u=Id$ we obtain a special case of the quantitative rigidity result of Friesecke, James and Muller. Our main result can be considered as a first step in a new line of generalization of F-J-M Theorem in which $Id$ is replaced by a mapping of non-trivial degree.