On the Injectivity of C¹ Maps of the Real Plane

Abstract

Abstract. Let X : R 2 → R 2 be a C 1 map. Denote by Spec(X) the set of (complex) eigenvalues of DX p when p varies in R 2 . If there exists ǫ > 0 such that Spec(X) ∩ (−ǫ, ǫ) = ∅, then X is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Jacobian Conjecture and semi-algebraic maps

“…In [13] Fernandes-GutierrezRabanal showed, for n = 2, that: If Spec(F) [0, ε) = 6, for some ε > 0, then F is injective. This result is an improvement of several results obtained previously by Gutierrez et al (see [7], [14] and [16]). As a consequence of this result the authors also prove the global asymptotic stability conjecture for differentiable vector fields of R 2 : If Spec(F) ⊂ {z ∈ C : Re(z) < 0}, then for all p in R 2 , there is a unique positive trajectory starting at p whose ω-limit set is exactly {0}.…”

Global asymptotic stability for differentiable vector fields of R2

“…(2) Theorem 2.1 implies the Fessler [13] and Gutierrez injectivity result, which requires X be of class C 1 and SpecðX Þ-½0; NÞ ¼ |: Theorem 2.1 also implies the Cobo et al [11] injectivity result, which needs X be of class C 1 and the existence of an e40 such that SpecðX Þ-ðÀe; eÞ ¼ |:…”

A new qualitative proof of a result on the real jacobian conjecture

“…if inverse images of compact subsets are compact), or F is a diffeomorphism if and only if These conditions are due to Banach-Mazur and Hadamard, respectively, and remain true in more general spaces, for details, see (Plastock 1974). Another condition, now specifically of R 2 and ensuring just the injectivity of F, is the following sufficient condition: the real eigenvalues of DF (x), for all x ∈ R 2 , are not contained in an interval of the form (0, ε), for some ε > 0, see (Fernandes et al 2004) and (Cobo et al 2002). Now, if F is a polynomial map, the statement that F is injective is known as the real Jacobian conjecture.…”