Image of iterated polynomial maps of the real plane (Q6191464)

The author proves the following theorem. Let \(F:\mathbb{R}^2\rightarrow \mathbb{R}^2\) be a polynomial mapping satisfying the following conditions: 1) The complement \(\mathbb{R}^2\setminus F(\mathbb{R}^2)\) consists of a finite number of points; 2) Each point outside the asymptotic variety of \(F\) has more than one preimage via \(F\). Then, there exists a natural number \(N\) such that the image \(F^k(\mathbb{R}^2)\) of \(\mathbb{R}^2\) under the \(k\)th iteration of \(F\) is the same for all \(k\geq N\).