Foliations and Polynomial Diffeomorphisms of ${\mathbb{R}}^3$

Abstract

Let $Y=(f,g,h):{\mathbb{R}}^3\to {\mathbb{R}}^3$ be a $C^2$ map and let $\spec(Y)$ denote the set of eigenvalues of the derivative $D{Y}_p$, when $p$ varies in ${\mathbb{R}}^3$. We begin proving that if, for some $\epsilon >0,$ $\spec(Y)\cap (-ε,ε)=\emptyset,$ then the foliation $\mathcal{F}(k),$ with $k\in \{f,g,h\},$ made up by the level surfaces $\{k=\mathrm{constant}\},$ consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek's Jacobian Conjecture for polynomial maps of ${\mathbb{R}}^n.$

13 pages and 3 figures