Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space

TL;DR

This paper extends conditions for injectivity of non-singular planar maps by replacing eigenvalue constraints with disconnecting curves in the eigenvalues space and explores Jacobian maps with non-surjective divergence.

Contribution

It generalizes previous injectivity results by using disconnecting curves in the eigenvalues space and characterizes injectivity via the non-surjectivity of the divergence of the Jacobian map.

Findings

Injectivity holds when eigenvalues lie outside unbounded disconnecting curves.

Injectivity is guaranteed if the divergence of the Jacobian map is not surjective.

Generalizes prior results on eigenvalue constraints for planar maps.

Abstract

Fessler and Gutierrez \cite{Fe,Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $P_x + Q_y$ is not a surjective function.