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

Carlos Gutierrez·Roland Rabanal
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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}.

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Carlos Gutierrez, Roland Rabanal. 2004-08-13. The Markus-Yamabe Conjecture for Differentiable vector fields of R^2. https://arxiv.org/abs/math/0311386

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