Global injectivity of C1 maps of the real plane, inseparable leaves and the palais-smale condition

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

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7 Citations (Scopus)

Abstract

We study two sufficient conditions that imply global injectivity for a C1 map X: ℝ2 → ℝ2 such that its Jacobian at any point of ℝ2 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. © Canadian Mathematical Society 2007.
Original languageEnglish
Pages (from-to)377-389
JournalCanadian Mathematical Bulletin
Volume50
DOIs
Publication statusPublished - 1 Jan 2007

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