On local diffeomorphisms of \(\mathbb{R}^n\) that are injective
Let \(F\) be a \(C^1\) map from \(\mathbb{R}^n\) to \(\mathbb{R}^n\) and let \(\text{Spec}(F)\) denote the eigenvalues of its derivative \(dF(p)\) as \(p\) ranges over \(\mathbb{R}^n\). The authors prove several theorems which guarantee that such a map \(F\) that is a local diffeomorphism be injective. They thereby obtain the equivalence of the weak Markus-Yamabe conjecture (if \(\text{Spec}(F) \subset \{ z \in \mathbb{C} : \operatorname{Re}(z) < 0 \}\) then \(F\) is injective) and the Chamber-Land conjecture (if \(\text{Spec}(F) \cap \{ z \in \mathbb{C} : | z| < \varepsilon \} = \emptyset\) for some \(\varepsilon > 0\) then \(F\) is injective) when \(F\) is Lipschitz. Other similar results are also established.
scientific article; zbMATH DE number 2164194
scientific article