A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$

Carlos Gutierrez; Nguyen Van Chau

doi:10.3934/dcds.2007.17.397

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2007, Volume 17, Issue 2: 397-402. Doi: 10.3934/dcds.2007.17.397

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A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$

Carlos Gutierrez^1, and
Nguyen Van Chau^2,

1.
Instituto de Ciências Matemáticas e de Computa¸cão - USP, Cx. Postal 668, CEP 13560–970, São Carlos, SP
2.
Institute of Mathematics, P.O. Box 1078, Hanoi

Received: December 2005

Revised: September 2006

Published: May 2007

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Abstract

Using the half-Reeb component technique as introduced in [10], we try to clarify the intrinsic relation between the injectivity of differentiable local homeomorphisms $X$ of $R^2$ and the asymptotic behavior of real eigen-values of derivations $DX(x)$. The main result shows that a differentiable local homeomorphism $X$ of $R^2$ is injective and that its image $X(R^2)$ is a convex set if $X$ satisfies the following condition: (*) There does not exist a sequence $R^2$ ∋ $x_i\rightarrow \infty$ such that $X(x_i)\rightarrow a\in \R^2$ and $DX(x_i)$ has a real eigenvalue $\lambda _i\rightarrow 0$. When the graph of $X$ is an algebraic set, this condition becomes a necessary and sufficient condition for $X$ to be a global diffeomorphism.

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Mathematics Subject Classification: Primary: 14R15; Secondary: 14E07, 14E09, 14E40.

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