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

摘要

We study two sufficient conditions that imply global injectivityfor a $C^1$ map $Xcolon R^2 o R^2$ such that its Jacobian at anypoint of $R^2$ is not zero. One is based on the notion ofhalf-Reeb component and the other on the Palais--Smale condition.We improve the first condition using the notion of inseparableleaves. We provide a new proof of the sufficiency of the secondcondition. We prove that both conditions are not equivalent, moreprecisely we show that the Palais--Smale condition implies thenonexistence of inseparable leaves, but the converse is not true.Finally, we show that the Palais--Smale condition it is not anecessary condition for the global injectivity of the map $X$.