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On the Injectivity of C1 Maps of the Real Plane

Published online by Cambridge University Press:  20 November 2018

Milton Cobo ,
Carlos Gutierrez  and
Jaume Llibre
Milton Cobo
Affiliation:
Departamento de Matemáticas, IBILCE-UNESP, São José do Rio Preto (SP), Brazil, e-mail: milton@mat.ibilce.unesp.br
Carlos Gutierrez
Affiliation:
Departamento de Matemáticas, IBILCE-UNESP, São José do Rio Preto (SP), Brazil, e-mail: milton@mat.ibilce.unesp.br
Jaume Llibre
Affiliation:
Departamento de Matemáticas, IBILCE-UNESP, São José do Rio Preto (SP), Brazil, e-mail: milton@mat.ibilce.unesp.br
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Abstract

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Let $X:\,{{\mathbb{R}}^{2}}\to \,{{\mathbb{R}}^{2}}$ be a ${{C}^{1}}$ map. Denote by $\text{Spec}(X)$ the set of (complex) eigenvalues of $\text{D}{{\text{X}}_{p}}$ when $p$ varies in ${{\mathbb{R}}^{2}}$. If there exists $\in \,>\,0$ such that $\text{Spec(}X)\,\bigcap \,(-\in ,\,\in )\,=\,\varnothing $, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Keywords

34D0554H2058F1058F21
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 6 , 01 December 2002 , pp. 1187 - 1201
Copyright
Copyright © Canadian Mathematical Society 2002

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