Global Injectivity of C 1 Maps of the Real Plane, Inseparable Leaves and the Palais–Smale Condition

Gutierrez, C. ; Jarque, X. ; Llibre, J. ; Teixeira, M. A.

Canadian mathematical bulletin, Tome 50 (2007) no. 3, pp. 377-389

Résumé

We study two sufficient conditions that imply global injectivity for a ${{C}^{1}}$ map $X:{{\mathbb{R}}^{2}}\to {{\mathbb{R}}^{2}}$ such that its Jacobian at any point of ${{\mathbb{R}}^{2}}$ is not zero. One is based on the notion of half-Reeb component and the other on the Palais–Smale condition. We improve the first condition using the notion of inseparable leaves. We provide a new proof of the sufficiency of the second condition. We prove that both conditions are not equivalent, more precisely we show that the Palais–Smale condition implies the nonexistence of inseparable leaves, but the converse is not true. Finally, we show that the Palais–Smale condition it is not a necessary condition for the global injectivity of the map $X$ .

Détail

DOI : 10.4153/CMB-2007-036-0

Mots-clés : 34C35, 34H05

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