Global injectivity of differentiable maps via W-condition in $\mathbb{R}^2$

Wei Liu; Wei Liu

doi:10.3934/math.2021097

AIMS Mathematics

2021, Volume 6, Issue 2: 1624-1633. doi: 10.3934/math.2021097

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Research article

Global injectivity of differentiable maps via W-condition in $\mathbb{R}^2$

Wei Liu ^,

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received: 13 September 2020 Accepted: 16 November 2020 Published: 25 November 2020
MSC : 14A25, 14R15, 26B10

In this paper, we study the intrinsic relations between the global injectivity of the differentiable local homeomorphism map $F$ and the rate of the Spec$(F)$ tending to zero, where Spec$(F)$ denotes the set of all (complex) eigenvalues of Jacobian matrix $JF(x)$, for all $x\in \mathbb{R}^2$. They depend deeply on the $W$-condition which extends the $*$-condition and the $B$-condition. The $W$-condition reveals the rate that tends to zero of the real eigenvalues of $JF$, which can not exceed $\displaystyle O\Big(x\ln x(\ln \frac{\ln x}{\ln\ln x})^2\Big)^{-1}$ by the half-Reeb component method. This improves the theorems of Gutiérrez ^[16] and Rabanal ^[27]. The $W$-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture.
- global injectivity,
- $W$-condition,
- half-Reeb component,
- Jacobian conjecture
Citation: Wei Liu. Global injectivity of differentiable maps via W-condition in $\mathbb{R}^2$[J]. AIMS Mathematics, 2021, 6(2): 1624-1633. doi: 10.3934/math.2021097

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Abstract

In this paper, we study the intrinsic relations between the global injectivity of the differentiable local homeomorphism map $F$ and the rate of the Spec$(F)$ tending to zero, where Spec$(F)$ denotes the set of all (complex) eigenvalues of Jacobian matrix $JF(x)$, for all $x\in \mathbb{R}^2$. They depend deeply on the $W$-condition which extends the $*$-condition and the $B$-condition. The $W$-condition reveals the rate that tends to zero of the real eigenvalues of $JF$, which can not exceed $\displaystyle O\Big(x\ln x(\ln \frac{\ln x}{\ln\ln x})^2\Big)^{-1}$ by the half-Reeb component method. This improves the theorems of Gutiérrez ^[16] and Rabanal ^[27]. The $W$-condition is optimal for the half-Reeb component method in this paper setting. This work is related to the Jacobian conjecture.

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