Injectivity of C1 maps R2→R2 at infinity and planar vector fields

Let X : R-2 \D-sigma --> R-2 be a C-1 map, where sigma > 0 and D-sigma = {p is an element of R-2 : parallel topparallel to less than or equal to sigma}. (i) If for some epsilon > 0 and for all p is an element of R-2 \ D-sigma, no eigenvalue of DX(p) belongs to (-epsilon, infinity), there exists s greater than or equal to sigma, such that X\(R2\Ds) is injective; (ii) If for some epsilon > 0 and for all p is an element of R-2 \ D-sigma, no eigenvalue of DX(p) belongs to (-epsilon, 0] boolean OR {z is an element of C : R(z) greater than or equal to 0}, there exists p(0) is an element of R-2 such that the point infinity, of the Riemann sphere R-2 boolean OR {infinity}, is either an attractor or a repellor of x' = X(x) + p(0).