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

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