Unipotent Jacobian matrices and univalent maps

Abstract

The Jacobian Conjecture would follow if it were known that real polynomial maps with a unipotent Jacobian matrix are injective. The conjecture that this is true even for C 1 maps is explored here. Some results known in the polynomial case are extended to the C 1 context, and some special cases are resolved.

On the Injectivity of C1 Maps of the Real Plane

“…Definition 3 Let f : R 2 → R be a C 1 submersion and let A and B be disjoint half-Reeb components for X f . We say that A and B are aligned and denote {A, B}, if there is a smooth embedded curve γ : [1,2] → R 2 such that the following properties are satisfied: (a3) A and B are both either on the left or on the right of the curve γ.…”

“…(4) Campbell [2] classified the two-dimensional C 1 maps whose eigenvalues are both 1. All such maps have an explicit inverse.…”

Global asymptotic stability for differentiable vector fields of R2

“…(4) Campbell [4] classified the two-dimensional C 1 maps whose eigenvalues are both 1: All such maps have an explicit inverse. The class of functions considered in Theorem 2.1 is much broader, but no explicit inverse is given.…”

“…Also, the surjectivity of the maps studied by Campbell remains as an open problem for the case of differentiable maps. The articles [8,10] are related to [4].…”

Injectivity and almost global asymptotic stability of Hurwitz vector fields

“…verifies that JH is nilpotent, rank(JH) = 2 and H 1 , H 2 , H 3 are linearly independent over κ. Moreover, L.A. Campbell in [4] generalizes this counterexample obtaining…”