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Global injectivity of maps and related topics

Abstract

Let F: R^n --> R^n be a local diffeomorphism. This project contains some directions toward determining additional conditions guaranteeing the global injectivity of F. The mechanisms of investigation are inside or are related to the area of dynamical systems, mainly concerning the qualitative theory of differential equations. We present three lines of research related to our projects along the last years with Brazilian and foreign collaborators. The first one deals with the polynomial case in R^2 and reach conditions related to the degree of F as well as the study of foliations of R^2 rising from polynomial submersions. The second one still deals with the polynomial case in R^2, but now under the viewpoint of the relations that the problem has with centers of polynomial Hamiltonian vector fields. Finally the third line of research deals with the n-dimensional case, with n greater than or equal to 3, as well as with some aspects of the injectivity problem in the complex case, i.e., when in place of R^n we put C^n. (AU)

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