On differentiable area-preserving maps of the plane (Q968025)

A map \(F:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}\) is an almost-area-preserving map if (a) \(F\) is an injective local homeomorphism, (b) there exists a constant \(s>0\) such that for every measurable set \(B,\) \( \mu (F(B))=s\mu (B).\) The main result is: if \(F\) is differentiable, \( \text{Jac}F\equiv c\neq 0\) (it implies, of course, that \(F\) is a local homeomorphism and satisfies (b)) and there does not exist a sequence \( (x_{k},y_{k})\in \mathbb{R}^{2}\) with \(x_{k}\rightarrow \infty \) such that \( \left( R_{\theta }\circ F\circ R_{-\theta }\right) (x_{k},y_{k})\rightarrow p\in \mathbb{R}^{2}\) (\(R_{\theta }\) is the rotation on the angle \(\theta\)) and \(D\left( R_{\theta }\circ F\circ R_{-\theta }\right) (x_{k},y_{k})\) has a real eigenvalue \( \lambda _{k}\) satisfying \(x_{k}\lambda _{k}\rightarrow 0\) then \(F\) is an almost-area-preserving map with convex image.