Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale Condition
Printed: Sep 2007
M. A. Teixeira
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 $X$.
34C35 - unknown classification 34C35
34H05 - Control problems [See also 49J15, 49K15, 93C15]