Injetividade global de difeomorfismos locais: alguns resultados com ênfase no caso polinomial

Abstract

The Real Jacobian Conjecture (RJC) states that a polynomial map of the plane in the plane with nonzero Jacobian at every point is invertible. Although we know that this conjecture is false, it is of great interest to find conditions that guarantee the global invertibility of local diffeomorphisms, in particular, of polynomial maps in the plane. In this dissertation, we explore results related to the RJC, with emphasis on results based on the Qualitative Theory of Ordinary Differential Equations. In particular, we explore the relation between global injectivity and topological equivalence of parallel vector fields.