The Markus-Yamabe Conjecture for Differentiable vector fields of R^2

Abstract

(a) Let X=(f,g) be a differentiable map in the plane (not necessarily C^1) and let Spec(X) be the set of (complex) eigenvalues of the derivative DX(p) when p varies in R^2. If, for some ε>0, the set Spec(X) is disjoint of [0,ε) then X is injective. (b) Let X be a differentiable vector field such that X(0)=0 and $Re(z)< 0$ for all z in Spec(X). Then, for all p in R^2, there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to {0}.