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

http://dx.doi.org/10.4153/CMB-2007-036-0
Canad. Math. Bull. 50(2007), 377-389

  Published:2007-09-01
 Printed: Sep 2007

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 $X$.