The Markus-Yamabe Conjecture for Differentiable vector fields of R²

2003/11/21 by Carlos Gutiérrez, Carlos Gutierrez, Gutierrez, Carlos +2
Mathematics · #26B10 #34D20 (Secondary) #34D23 #34D45 (Primary) #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Holomorphic and Operator Theory #Nonlinear Differential Equations Analysis #math.DS #msc:26B10 #msc:34D20 #msc:34D23 #msc:34D45

paper · pdf · doi:10.48550/arxiv.math/0311386

ABSTRACT

openalex publication_date 2003/11/21 · arxiv created 2004/08/13 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

(a) Let X=(f,g) be a differentiable map in the plane (not necessarily C¹) and let Spec(X) be the set of (complex) eigenvalues of the derivative DX(p) when p varies in R². If, for some ε>0, the set Spec(X) is disjoint of [0,ε) then X is injective. (b) Let X be a differentiable vector field such that X(0)=0 and Re(z)< 0 for all z in Spec(X). Then, for all p in R², there is a unique positive trajectory starting at p; moreover the ω-limit set of p is equal to 0.

Related

• Injectivity of differentiable maps R² --> R² at infinity
• Asymptotic stability at infinity for differentiable vector fields of the plane
• The Discrete and Continuous Markus-Yamabe Stability Conjectures
• Global injectivity of differentiable maps via W-condition in R²
• An extension to the planar Markus-Yamabe Jacobian conjecture