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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: [email protected]
Carlos Gutierrez
Affiliation:
Departamento de Matemáticas, IBILCE-UNESP, São José do Rio Preto (SP), Brazil, e-mail: [email protected]
Jaume Llibre
Affiliation:
Departamento de Matemáticas, IBILCE-UNESP, São José do Rio Preto (SP), Brazil, e-mail: [email protected]
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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

References

[1] Bialynicki-Birula, A. and Rosenlicht, M., Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc. 13 (1962), 200–203.Google Scholar
[2] Campbell, L. A., Unipotent Jacobian matrices and univalent maps. Contemp.Math. 264 (2000), 157–177.Google Scholar
[3] Chamberland, M. and Meisters, G., A mountain pass to the Jacobian Conjecture. Canad. Math. Bull. 41 (1998), 442–451.Google Scholar
[4] Dumortier, F., Singularities of vector fields on the plane. J. Differential Equations 23, 1 (1977), 53–106.Google Scholar
[5] van den Essen, A., Polynomial automorphisms and the Jacobian conjecture. Progr. Math. 190, Birkhauser Verlag, Basel, 2000.Google Scholar
[6] Fessler, R., A proof of the two dimensional Markus-Yamabe Stability Conjecture and a generalization. Ann. Polon. Math. LXII(1995), 45–74.Google Scholar
[7] Golubitsky, M. and Guillemin, V., Stable mapings and their singularities. Grad. Texts in Math. 14, Springer-Verlag, 1973.Google Scholar
[8] González, E. A., Generic properties of polynomial vector fields at infinity. Trans. Amer.Math. Soc. 143 (1969), 201–222.Google Scholar
[9] Gutierrez, C., Smoothability of Cherry flows on two-manifolds. Lecture Notes in Math. 1007 (1981), 308–331.Google Scholar
[10] Gutierrez, C., A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. Henri Poincaré 12 (1995), 627–671.Google Scholar
[11] Kaplan, W., Regular curve-families filling the plane II. Duke Math. J. 8 (1941), 11–46.Google Scholar
[12] Kurdyka, K., Injective endomorphisms of real algebraic sets are surjective. Math. Ann. 313 (1999), 69–82.Google Scholar
[13] Markus, L. and Yamabe, H., Global stability criteria for differential systems. Osaka Math. J. 12 (1960), 305–317.Google Scholar
[14] Newman, D. J., One-to-one polynomial maps. Proc. Amer.Math. Soc. 11 (1960), 867–870.Google Scholar
[15] Pinchuck, S., A counterexample to the strong Jacobian conjecture.Math. Z. 217 (1994), 1–4.Google Scholar