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

http://dx.doi.org/10.4153/CJM-2002-045-0
Canad. J. Math. 54(2002), 1187-1201

  Published:2002-12-01
 Printed: Dec 2002

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.