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# On the Injectivity of $C^1$ Maps of the Real Plane

Published:2002-12-01
Printed: Dec 2002
• Milton Cobo
• Carlos Gutierrez
• Jaume Llibre
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## Abstract

Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of (complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists $\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.
 MSC Classifications: 34D05 - Asymptotic properties 54H20 - Topological dynamics [See also 28Dxx, 37Bxx] 58F10 - unknown classification 58F1058F21 - unknown classification 58F21

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