A178/2002 - Injectivity of C^1 maps R^2 -> R^2 at infinity and planar vector fields

Preprint A178/2002

Injectivity of C^1 maps R^2 -> R^2 at infinity and planar vector fields

Alberto Sarmiento | Gutierrez, Carlos

Keywords: planar vector fields | global injectivity

Let $X:\R^2\setminus \overline{D}_\sigma \to \R^2$ be a $C^1$ map, where $\sigma>0$ and $\overline{D}_\sigma = \{ p\in \R^2 : \vert\vert p \vert\vert \le \sigma\}.$ \\ (i) If for some $\epsilon >0$ and for all $p\in \R^2\setminus \overline{D}_\sigma,$ no eigenvalue of $DX(p)$ belongs to $(-\epsilon, \infty),$ there exists $s \ge \sigma$, such that $X|_{\R^2\setminus \overline{D}_s}$ is injective; \\ (ii) If for some $\epsilon >0$ and for all $p\in \R^2\setminus \overline{D}_\sigma,$ no eigenvalue of $DX(p)$ belongs to $(-\epsilon, 0]\cup \{ z\in\C: \Re(z)\ge 0 \},$ there exists $p_0\in \R^2$ such that the point $\infty$, of the Riemann sphere $\R^2\cup \{\infty\},$ is either an attractor or a repellor of $x'= X(x) + p_0.$