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# Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition

Published:2007-09-01
Printed: Sep 2007
• C. Gutierrez
• X. Jarque
• J. Llibre
• M. A. Teixeira
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## Abstract

We study two sufficient conditions that imply global injectivity for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any point of $\R^2$ 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 $X$.
 MSC Classifications: 34C35 - unknown classification 34C3534H05 - Control problems [See also 49J15, 49K15, 93C15]

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