Abstract

Let $X:U\to R2$ be a vector field defined on the complement of a compact set. We study the intrinsic relation between the asymptotic behavior of the real eigenvalues of the differential $DX$_z and the global injectivity of the local diffeomorphism given by $X$. The set $U$ induces a neighborhood of $\infty$ in the Riemann Sphere $R2\cup \{\infty \}$. In this talk, we speak about the existence of a sufficient condition that imply that the vector field $X:(U,\infty )\to (R2,\; 0)$, which is differentiable in $U\backslash \{\infty \}$ but not necessarily continuous at $\infty$, has $\infty$ as an attracting or a repelling singularity.

This improves the main result of Gutierrez-Sarmiento (2003) Asterisque, 287, 89-102.