An eigenvalue condition for the injectivity and asymptotic stability at infinity

Let X: U → ℝ² be a differentiable 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. This set U induces a neighborhood of ∞ in the Riemann Sphere ℝ² ∪ {∞}. In this work we prove the existence of a sufficient condition which implies that the vector field X: (U,∞) → (ℝ², 0), -which is differentiable in U \{∞} but not necessarily continuous at ∞,- has ∞ as an attracting or a repelling singularity. This improves the main result of Gutiérrez-Sarmiento: Asterisque, 287 (2003) 89-102.