Image of iterated polynomial maps of the real plane

Let \(F: {\mathbb {R}}^2\rightarrow {\mathbb {R}}^2\) $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be a polynomial mapping. We consider the image of the compositions \(F^k\) $F^k$ of F. We prove that under some condition then the image of the iterated map \(F^k\) $F^k$ is stable when k is large.