On local diffeomorphisms of \(\mathbb{R}^n\) that are injective
type: Item
instance of: scholarly article
title: On local diffeomorphisms of \(\mathbb{R}^n\) that are injective
zbMATH Open document ID: 1067.37019
DOI: 10.1007/BF02970861
author: Niranjan Chakravarthy
published in: Qualitative Theory of Dynamical Systems
publication date: 2005-05-03T00:00:00Z
review text: 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.
reviewed by: Douglas S. Shafer
Mathematics Subject Classification ID: 37C05
zbMATH DE Number: 2164194
zbMATH Keywords: diffeomorphism
zbMATH Keywords: injectivity
zbMATH Keywords: weak Markus-Yamabe conjecture
zbMATH Keywords: Chamber-Land conjecture
Wikidata QID: Q114233713
MaRDI profile type: Publication
cites work: The Jacobian conjecture: Reduction of degree and formal expansion of the inverse
cites work: Q4146134
cites work: Unipotent Jacobian matrices and univalent maps
cites work: Characteristic values of the Jacobian matrix and global invertibility
cites work: A Mountain Pass to the Jacobian Conjecture
cites work: Global inversion via the Palais-Smale condition
cites work: A counterexample to the strong real Jacobian conjecture
cites work: Homeomorphisms Between Banach Spaces
cites work: Injectivity of local diffeomorphisms from nearly spectral conditions
cites work: Polynomial automorphisms and the Jacobian conjecture
cites work: Two remarks on the stability of ordinary differential equations
cites work: Keller's problem
Recommended article: Injectivity of local diffeomorphisms from nearly spectral conditions
Recommended article: Locally injective automorphic mappings in $R^n$
Recommended article: Injectivity of polynomial local homeomorphisms of Rn
Recommended article: Q4463775
Recommended article: Q3831689
Recommended article: Local homeo- and diffeomorphisms: invertibility and convex image
Recommended article: Fibrations and global injectivity of local homeomorphisms
Recommended article: Foliations and global injectivity in \(\mathbb R^{n }\)
Recommended article: Q4302625
Recommended article: A new class of non-injective polynomial local diffeomorphisms on the plane
description: scientific article; zbMATH DE number 2164194