Global injectivity of differentiable maps via W-condition in R^2

TL;DR

This paper establishes a new condition called the W-condition that characterizes the global injectivity of differentiable maps in , linking eigenvalue decay rates to injectivity and improving previous theorems.

Contribution

The paper introduces the W-condition, extending previous conditions, and demonstrates its optimality for ensuring global injectivity of differentiable maps in .

Findings

The W-condition relates eigenvalue decay to injectivity.

Eigenvalues cannot decay faster than a specific rate under the W-condition.

The W-condition improves upon previous injectivity criteria.

Abstract

In this paper, we study the intrinsic relation between the global injectivity of differentiable local homeomorphisms $F$ and the rate that tends to zero of $Spec(F)$ in $\mathbb{R}^2$, where $Spec(F)$ denotes the set of all (complex) eigenvalues of $DF(x)$, for all $x\in \mathbb{R}^2$. This depends on the $W$-condition deeply, which extends the $*$-condition and $B$-condition. The $W$-condition reveals the rate that tends to zero of real eigenvalues of $DF$ 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 Guti\'{e}rrez-Nguyen \cite{GN07} and Rabanal \cite{RR10}. The $W$-condition is optimal for the half-Reeb component method in this paper setting.

Equations

$$Spec(X)\cap [0,\epsilon )=\varnothing ,$$

$$R_{\theta }=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

$$0\notin Spec(F)\text{mbox}andSpec(F+F^T)\subseteq (-\infty ,-\epsilon )\text{mbox}or(\epsilon ,+\infty ),$$

$$\frac{1}{x_k\ln x_k}={\lambda }_k\to 0.$$

$$B=\{(x,y)\in[0,2]\times[0,2]\big{|}0<x+y\leqslant 2\}.$$

$$(x,y)\to (x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta ),$$

$$F_{\theta }:=(f_{\theta },g_{\theta })=R_{\theta }\circ F\circ R_{-\theta }.$$

$$H(x)=\sup \{y:(x,y)\in {\Pi }^{-1}(x)\cap {\alpha }_x\}.$$

$$\Phi:(a,+\infty)~{}~{}\mbox{ by}~{}~{}\Phi(x)=f_{\theta}\big{(}x,H(x)\big{)}.$$

$$DF_{\theta}(x,H(x))=\begin{pmatrix}\Phi^{\prime}(x)&0\\ \partial_{x}g_{\theta}\big{(}x,H(x)\big{)}&\partial_{y}g_{\theta}\big{(}x,H(x)\big{)}\end{pmatrix}.$$

$$\underset{x_k\to +\infty }{\mathrm{liminf}}{\Phi }^{\prime }(x_k)P(x_k)>0.$$

$${\Phi }^{\prime }(x)P(x)>{\epsilon }_0,\forall x⩾a_0.$$

$$\Phi (x)-\Phi (a_0)⩽L,\forall x⩾a_0.$$

$${\int }_{a_0}^C\frac{1}{P(x)}dx>\frac{L}{{\epsilon }_0}.$$

$$L⩾\Phi (C)-\Phi (a_0)={\int }_{a_0}^C{\Phi }^{\prime }(x)dx⩾{\int }_{a_0}^C\frac{{\epsilon }_0}{P(x)}dx>L.$$

$$\displaystyle\int_{\Omega}{\big{(}1-\det F^{\prime}(x)\big{)}dx}=0,~{}\forall~{}\Omega\subset{\mathbb{R}^{n}}.$$

$${\int }_{U\left(x_0,\delta \right)}(1-\det F^{\prime }(x))dx\le {\int }_{U\left(x_0,\delta \right)}-\frac{C}{2}dx=-\frac{C}{2}m(U\left(x_0,\delta \right))<0,$$

$$DF(x,H(x))=\begin{pmatrix}\Phi^{\prime}(x)&0\\ \partial_{x}g\big{(}x,H(x)\big{)}&\partial_{y}g\big{(}x,H(x)\big{)}\end{pmatrix}.$$

$${\int }_{C_0}^C\frac{l}{P(x)}dx>K.$$

$$K<{\int }_{C_0}^C\frac{l}{P(x)}dx\le {\int }_{C_0}^C{\Phi }^{\prime }(x)dx<K.$$

Full text

Global injectivity of differentiable maps via W-condition in $\mathbb{R}^{2}$  ††thanks: Supported by NSFC.  E-mails: liuw16@mails.tsinghua.edu.cn

Wei Liu

*Department of Mathematical Sciences, Tsinghua University,

Beijing 100084, China*

Abstract

In this paper, we study the intrinsic relation between the global injectivity of differentiable local homeomorphisms $F$ and the rate that tends to zero of $Spec(F)$ in $\mathbb{R}^{2}$, where $Spec(F)$ denotes the set of all (complex) eigenvalues of $DF(x)$, for all $x\in\mathbb{R}^{2}$. This depends on the $W$-condition deeply, which extends the $*$-condition and $B$-condition. The $W$-condition reveals the rate that tends to zero of real eigenvalues of $DF$ 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-Nguyen [15] and Rabanal [24]. The $W$-condition is optimal for the half-Reeb component method in this paper setting.

Key words: $W$-condition; Half-Reeb component; Jacobian conjecture.

MSC(2010): 14R15; 14E07; 14E09.

1 Introduction

On the long-standing Jacobian conjecture, it is still open even in the case $n=2$. There are many results on it, see for example [1] and [8].

A very important step, for example in $\mathbb{R}^{2}$, is the following result, due to A. Fernandes, C. Gutiérrez, and R. Rabanal:

Theorem 1.1**.**

([9]) Let $X=(f,g):\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ be a differentiable map. For some $\varepsilon>0$, if

[TABLE]

then $X$ is injective.

Theorem 1.1 is deep. If the assumptions (1.1) replaced by $0\notin$ Spec $(F)$, then the conclusion is false, even for polynomial map $X$, as the Pinchuck’s counterexample[P94]. Pmyth and Xavier[27] proved that there exists $n>2$ and non-injective polynomial map such that Spec$(X)\cap[0,+\infty)=\varnothing.$

Theorem 1.1 added to a long sequence of results on Markus-Yamabe conjecture[21] and the eigenvalue conditions of some map for injectivity in dimension two. The Markus-Yamabe Conjecture has been solved independently in 1993 by C. Gutierrez[12] and R. Fessler[11]. It is false in dimension $n\geqslant 3$ even for polynomial vector field[6]. Theorem 1.1 also implies that the following conjecture is true in dimension $n=2$.

Conjecture 1.1**.**

([4], Conjecture 2.1) Let $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a ${C^{1}}$ map. Suppose there exists an $\varepsilon>0$ such that $|\lambda|\geqslant\varepsilon$  for all the eigenvalues $\lambda$ of $F^{\prime}(x)$ and all $x\in\mathbb{R}^{n}$. Then $F$ is injective.

The essential tool to prove Theorem 1.1 is making use of the concept of the half-Reeb component that we recall in Definition 2.1.

C. Gutiérrez and V. Ch. Nguyen [15] study the geometrical behavior of differentiable maps in $\mathbb{R}^{2}$ and the following $*$-condition on the real eigenvalues of $DF$ under the half-Reeb component technique.

For each $\theta\in\mathbb{R}$, we denote by $R_{\theta}$ the linear rotation

[TABLE]

and define the map $F_{\theta}=R_{\theta}\circ F\circ R_{-\theta}$.

Definition 1.1**.**

([15], $*$-condition) A differentiable $F$ satisfies the $*$-condition if for each $\theta\in\mathbb{R}$, there does not exist a sequence $\mathbb{R}^{2}\ni z_{k}\to\infty$ such that, $F_{\theta}(z_{k})\to T\in\mathbb{R}^{2}$ and $DF_{\theta}(z_{k})$ has a real eigenvalue $\lambda_{k}\to 0$.

Theorem 1.2**.**

([15]) Suppose that $X:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ is a differentiable local homeomorphism. Then:

(i) If $X$ satisfies $*$-condition, then $X$ is injective and its image is a convex set.

(ii) $X$ is a global homeomorphism of $\mathbb{R}^{2}$ if and only if $X$ satisfies $*$-condition and its image $X(\mathbb{R}^{2})$ is dense in $\mathbb{R}^{2}$.

$*$-condition is somewhat weaker than condition (1.1), thus one can obtain the Theorem 1.1 from Theorem 1.2 (i) by a standard procedure.

In other new case, the essential difficulty is that the eigenvalues of $DF$ which may be tending to zero implies $F$ is globally injective. R. Rabanal[24] extended the $*$ condition to the so called $B$-condition.

Definition 1.2**.**

([24], $B$-condition) The differentiable map $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ satisfies the $B$-condition if for each $\theta\in\mathbb{R}$, there does not exist a sequence $(x_{k},y_{k})\in\mathbb{R}^{2}$ with $x_{k}\to+\infty$ such that $F_{\theta}(x_{k},y_{k})\to T\in\mathbb{R}^{2}$ and $DF_{\theta}(x_{k},y_{k})$ has a real eigenvalue $\lambda_{k}$ satisfying $\lambda_{k}x_{k}\to 0.$

He obtains the following theorem where Theorem 1.2 holds if one replaced $*$-condition by $B$-conditon.

Theorem 1.3**.**

([24]) Suppose that the differentiable map $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ satisfies the $B$-condition and $\det DF(z)\neq 0,\forall z\in\mathbb{R}^{2}$, then $F$ is a topological embedding.

In fact, Theorem 1.3 improves the main results of [15], (see [26],[23]).

In 2014, F. Braun and V. S. Jean[3] considered the relation between the half-Reeb component and Palais-Smale condition for global injectivity.

Many references on other aspects of half Reeb component including in higher dimensional situations see ([25], [18],[19], [20], [13]).

For example, C. Gutiérrez and C. Maquera considered half-Reeb components for the global injectivity in dimension 3.

Theorem 1.4**.**

([13]) Let $Y=(f,g,h):\mathbb{R}^{3}\to\mathbb{R}^{3}$ be a polynomial map such that $Spec(Y)\cap[0,\varepsilon)=\emptyset$, for some $\varepsilon>0$. If $codim(SY)\geqslant 2$, then $Y$ is a bijection.

Recently, W. Liu prove the following theorem by the Minimax method.

Theorem 1.5**.**

([19]) Let $F:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a $C^{1}$ map, $n\geqslant 2$. If for some $\varepsilon>0$,

[TABLE]

then $F$ is globally injective.

Let us return to study approaching to zero of the eigenvalues of $DF$ by the half-Reeb component method in $\mathbb{R}^{2}$.

In this paper, we define the $W$-condition and obtain the following result.

For the convenience of our statement, let us set

\mathcal{P}=\Big{\{}P~{}\big{|}~{}\mathbb{R}^{+}\to\mathbb{R}^{+},P~{}\mbox{is nondecreasing and}~{}\forall M>0,\mbox{there exists large constant} $N$ which depends on $M$ and $P$, such that \displaystyle\int_{2}^{N}\frac{1}{P(x)}dx>M\Big{\}}.

Thus, $\mathcal{P}$ contains many functions, such as 1, $x$, $x\ln(x+1)$, x\ln(1+x)\ln\big{(}1+\ln(1+x)\big{)} and it does not include $x^{\alpha},\forall\alpha>1;~{}x\ln^{\beta}(x+1),\forall\beta>1.$

Definition 1.3**.**

($W$-condition)

A differentiable map $F$ satisfies the $W$-condition if for each $\theta\in\mathbb{R}$ (see (1.2)), there does not exist a sequence $(x_{k},y_{k})\in\mathbb{R}^{2}$ with $x_{k}\to+\infty$ such that $F_{\theta}(x_{k},y_{k})\to T\in\mathbb{R}^{2}$ and $DF_{\theta}(x_{k},y_{k})$ has a real eigenvalue $\lambda_{k}$ satisfying $\lambda_{k}P(x_{k})\to 0$, where $P\in\mathcal{P}$.

Remark 1.1**.**

$W$-condition obviously contains $*$-condition and $B$-condition. Let $P(x)=x\ln(x+1)\in\mathcal{P}$, the $W$-condition with the $P$ is weaker than $*$-condition and $B$-condition. It seems can not be improved in this setting by making use of the half-Reeb component method. The $W$-condition profoundly reveals the optimal rate that tends to zero of eigenvalues of $DF$ must be in the interval \displaystyle\Big{(}O(x\ln^{\beta}x)^{-1},\forall\beta>1, O\Big{(}x\ln x\big{(}\ln\frac{\ln x}{\ln\ln x}\big{)}^{2}\Big{)}^{-1}\Big{]} by the half-Reeb component method.

Remark 1.2**.**

If $x_{k}$ exchanges $y_{k}$ in definition 1.3, then it is also applied.

Remark 1.3**.**

For example, let $g(x,y)$ is a $C^{1}$ function such that $g(x,y)=\frac{y}{x\ln x}$ where $x\geqslant 2$. The map $F(x,y)=(e^{-x},g(x,y))$ satisfies $\det JF=-e^{-x}\frac{1}{x\ln x}\neq 0$. Then, for $\{x_{k}\}\subseteq[2,+\infty)$, $F(x_{k},0)=(e^{-x_{k}},0)\to P=(0,0)$, as $x_{k}\to+\infty$. $JF(x_{k},0)$ has a real eigenvalue

[TABLE]

However, the limit of the product $x_{k}\ln x_{k}$ is away from zero.

We use the $W$-condition and obtain the following results.

Theorem 1.6**.**

Let $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a differentiable local homeomorphism. If $F$ satisfies $W$-condition, then $F$ is injective and $F(\mathbb{R}^{2})$ is convex.

Obviously, Theorem 1.6 implies Theorem 1.2 and Theorem 1.3(i).

Next, we also have the following results.

Theorem 1.7**.**

Let $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a differentiable Jacobian map. If $F$ satisfies $W$-condition, then $F$ is a globally injective, measure-preserving map with convex image.

The Theorem 1.7 improves the main results of GN [15] and Ra [24].

Since the map is injective in Theorem in 1.7, we obtain some fixed point theorem, that’s the following corollary.

Corollary 1.1**.**

If $F$ is as in Theorem 1.7 and Spec(F)\subseteq\{z\in\mathbb{C}\big{|}|z|<1\} , then $F$ has at most one fixed point.

Another important property on the Keller maps as in corollary 1.1 is theroem $B$ in [5]. It proves that a global attractor for the discrete dynamical system has a unique fixed point.

By the Inverse Function Theorem, the $F$ in Theorem 1.6 is locally injective at any point in $\mathbb{R}^{2}$. However, in general, it is not global injective map. So the goal is to give the sufficient conditions in order to get the global injectivity of $F$. We use the $W$ condition to get the following theorem.

Theorem 1.8**.**

Let $F=(f,g):\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a local homeomorphism such that for some $s>0,~{}~{}F|_{\mathbb{R}^{2}\backslash D_{s}}$ is differentiable. If $F$ satisfies the $W$-condition, then it is a globally injective and $F(\mathbb{R}^{2})$ is a convex set.

Remark 1.4**.**

If the graph of $F$ is an algebraic set , then the injectivity of $F$ must be the bijectivity of $F$.

$W$ condition can be also devoted to study the differentiable map $F:{\mathbb{R}^{2}\backslash D_{s}}\to\mathbb{R}^{2}$ whose the $Spec(F)$ is disjoint with $[0,+\infty)$.

Theorem 1.9**.**

Let $F=(f,g):\mathbb{R}^{2}\backslash\overline{D_{\sigma}}\to\mathbb{R}^{2}$ be a differential map which satisfies the $W$-condition. If $Spec(F)\cap[0,+\infty)=\emptyset$ or $Spec(F)\cap(-\infty,0]=\emptyset$, then there exists $s\geqslant\sigma$ such that $F|_{\mathbb{R}^{2}\backslash D_{s}}$ can be extended to an injective local homeomorphism $\widetilde{F}=(\widetilde{f},\widetilde{g}):\mathbb{R}^{2}\to\mathbb{R}^{2}$.

These works are related to the Jacobian conjecture which can be reduce to that for all dimension $n\geqslant 2$, a polynomial map $F:\mathbb{C}^{n}\to\mathbb{C}^{n}$ of the form $F=x+H$, where $H$ is cube-homogeneous and $JH$ is symmetry, is injective if $Spec(F)=\{1\}$. (see [2]).

In order to prove our theorems, we need to use the definition and propositions of the half-Reeb component.

2 Half-Reeb component

In this section, we will introduce some preparation on the eigenvalue conditions of $Spec(F)$.

Let $h_{0}(x,y)=xy$ and consider the set

[TABLE]

Definition 2.1**.**

(half-Reeb component[12]) Let $X$ be a differentiable map from $\mathbb{R}^{2}\to\mathbb{R}^{2}$. $\det DX_{p}\neq,\forall p\in\mathbb{R}^{2}$, Given $h\in\{f,g\}$, we will say that $\mathcal{A}\subseteq\mathbb{R}^{2}$ is a half-Reeb component for $\mathcal{F}(h)$ (or simply a hRc for $\mathcal{F}(h)$)if there exists a homeomorphism $H:B\to\mathcal{A}$ which is a topological equivalence between $\mathcal{F}(h)|_{\mathcal{A}}$ and $\mathcal{F}(h_{0})|_{B}$ and such that:

(1) The segment$\{(x,y)\in B:x+y=2\}$ is sent by $H$ onto a transversal section for the foliation $\mathcal{F}(h)$ in the complement of $H(1,1)$; this section is called the compact edge of $\mathcal{A}$;

(2) Both segments $\{(x,y)\in B:x=0\}$ and $\{(x,y)\in B:y=0\}$ are sent by $H$ onto full half-trajectories of $\mathcal{F}(h)$. These two semi-trajectories of $\mathcal{F}(h)$ are called the noncompact edges of $\mathcal{A}$.

Proposition 2.1**.**

[9]** Suppose that $X=(f,g):\mathbb{R}^{2}\in\mathbb{R}^{2}$ is a differentiable map such that $0\notin Spec(X)$. If $X$ is not injective, then both $\mathcal{F}(f)$ and $\mathcal{F}(g)$ have half-Reeb components.

Proposition 2.2**.**

[9]** Let $X=(f,g):\mathbb{R}^{2}\in\mathbb{R}^{2}$ be a non-injective, differentiable map such that $0\notin Spec(X)$: Let $\mathcal{A}$ be a hRc of $\mathcal{F}(f)$ and let $(f_{\theta},g_{\theta})=R_{\theta}\circ X\circ R_{-\theta}$, where $\theta\in\mathbb{R}$ and $R_{\theta}$ is in (1.2). If $\Pi(x,y)=x$ is bounded, where $\Pi:\mathbb{R}^{2}\to\mathbb{R}$ is given by $\Pi(x,y)=x$, then there is an $\varepsilon>0$ such that, for all $\theta\in(-\varepsilon,0)\cup(0,\varepsilon)$; $\mathcal{F}(f_{\theta})$ has a hRc $\mathcal{A}_{\theta}$ such that $\Pi(\mathcal{A}_{\theta})$is an interval of infinite length.

3 Half-Reeb component and $W$-condition

In this section, we will establish the essential fact that the $W$-condition ensures non-existence of half-Reeb component.

Let $F=(f,g):\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a local homeomorphism of $\mathbb{R}^{2}$. For each $\theta\in\mathbb{R}$, we denoted by $R_{\theta}$ the linear rotation \big{(}see (1.2)\big{)}:

[TABLE]

and

[TABLE]

In other words, $F_{\theta}$ represents the linear rotation $R_{\theta}$ in the linear coordinates of $\mathbb{R}^{2}.$

Proposition 3.1**.**

A differentiable local homeomorphism $F:\mathbb{R}^{2}\to\mathbb{R}^{2}$ which satisfies $W$-condition has no half-Reeb components.

Proof.

Suppose by contradiction that $F$ has a half-Reeb component. In order to obtain this result, we consider the map $(f_{\theta},g_{\theta})=F_{\theta}$. From Proposition 2.2, there exists some $\theta\in\mathbb{R}$, such that $\mathcal{F}(\mathcal{A}_{\theta})$ has a half-Reeb component which $\Pi(\mathcal{A})$ is unbounded interval, where $\Pi(\mathcal{A})$ denote orthgonal projection onto the first coordinate in $\mathcal{A}$. Therefore $\exists$ $b$ and a half-Reeb component $A$, such that $[b,+\infty)\subseteq\Pi(\mathcal{A})$. Then, for large enough $a>b$ and any $x\geqslant a$, the vertical line $\Pi^{-1}(x)$ intersects exactly the one trajectory $\alpha_{x}\cap[x,+\infty)=x$, i.e. $x$ is maximum of the the trajectory $\Pi_{\alpha_{x}}$. If $x\geqslant a$, the intersection $\alpha_{x}\cap\Pi^{-1}(x)$ is compact subset in $\mathcal{A}$.

Thus, we can define functions $H:(a,+\infty)\to\mathbb{R}$ by

[TABLE]

As $\mathcal{F}(f_{\theta})$ is a foliation and can obtain

[TABLE]

We can know that $\Phi$ is a bounded, strictly monotone function such that, for some full measure subset $M\subseteq(a,+\infty)$.

Since the image of $\Phi$ is contained in $f_{\theta}(\Gamma)$ where $\Gamma$ is compact edge of hRc $\mathcal{A}$, the function $\Phi$ is bounded in $(a,+\infty)$. Furthermore, $\Phi$ is continuous because $\mathcal{F}(f_{\theta})$ is a $C^{0}$ foliation. And since $\mathcal{F}(f_{\theta})$ is transversal to $\Gamma$, $\Phi$ is monotone strictly.

For the measure subset $M\subseteq(a,+\infty)$, such that $\Phi(x)$ is differentiable on $M$ and the Jacobian matrix of $F_{\theta}(x,y)$ at \big{(}x,H(x)\big{)} is

[TABLE]

Therefore, $\forall x\in M$, \Phi^{\prime}(x)=\partial_{x}f_{\theta}\big{(}x,H(x)\big{)} is a real eigenvalue of $DF_{\theta}(x,H(x))$ and we denote it by $\lambda(x):=\Phi^{\prime}(x)$.

Since $F$ is local homeomorphism, without loss of generality, we assume $\Phi$ is strictly monotone increasing, $\Phi^{\prime}(x)>0,\forall x\in M$. Let any function $P\in\mathcal{P}$, where

\mathcal{P}=\Big{\{}P~{}\big{|}~{}\mathbb{R}^{+}\to\mathbb{R}^{+},P~{}\mbox{is nondecreasing and}~{}\forall M>0,\mbox{there exists large constant } $N$ which depends on $M$ and $P$, such that \displaystyle\int_{2}^{N}\frac{1}{P(x)}dx>M\Big{\}}.

Claim:

[TABLE]

Because $P(x)$ and $\Phi^{\prime}(x)$ are both positive, we can suppose by contradition that

$\liminf_{x_{k}\rightarrow+\infty}\Phi^{\prime}(x_{k})P(x_{k})=0.$ There exists a subsequence denoted still $\{x_{k}\}$ with $x_{k}\to+\infty$ such that $\Phi^{\prime}(x_{k})P(x_{k})\to 0.$ That is $\lambda(x_{k})P(x_{k})\to 0.$ Since $F_{\theta}(\mathcal{A})$ is bounded, F_{\theta}\big{(}x_{k},H(x_{k})\big{)} converges to a finite value $T$ on compact set $\overline{\mathcal{F}_{\theta}(\mathcal{A})}$. This contradicts the $W$-condition.

Therefore, there exist constant $a_{0}~{}(a_{0}>2)$ and small $\varepsilon_{0}>0$, such that

[TABLE]

Since $\Phi(x)$ is bounded, there exists $L>0$, such that

[TABLE]

By the definiton of $\mathcal{P}$, we can choose $C$ large enough, such that

[TABLE]

Thus,

[TABLE]

It is contradiction.

∎

4 The Proof of Theorem 1.6

Proof.

Suppose by contradiction that $F$ is not injective. By Proposition 2.1, $F$ has a half-Reeb component, this contradicts Proposition 3.1 that $F$ has no half-Reeb component if $F$ satisfies the $W$-condition. ∎

5 The Proof of Theorem 1.7

Proof.

Firstly, we prove the equivalence of the differential Jacobian map and measure-preserving in any dimension $n$.

For any nonempty measurable set $\Omega\subset\mathbb{R}^{n}.$ Since $F:\mathbb{R}^{n}\to\mathbb{R}^{n},$

denote $V=\{F(x)\left|{x\in}\right.\Omega\}$. Let the components of $F(x)$ be $v_{i}(i=1,2...n)$, i.e. $F({x_{1}},...{x_{n}})=({v_{1}}({x_{1},...x_{n}}),...{v_{n}}({x_{1}...x_{n}})).$ So $dv=\det F^{\prime}(x)dx.$ Since $\det F^{\prime}(x)\equiv 1$, we get $dv=dx.$

Therefore, $\int_{V}{dv}=\int_{\Omega}{dx}.$ It implies $F$ preserves measure.

Inversely, let $v=F(x),~{}\forall x\in\Omega.$ We still denote $V=\{F(x)\left|{x\in}\right.\Omega\}.$

Since $F$ preserves measure, one gets $\int_{V}{dv}=\int_{\Omega}{dx}.$

Combining with $dv=\det F^{\prime}(x)dx,$ we obtain $\int_{V}{dv}=\int_{\Omega}{\det F^{\prime}(x)dx}.$

Thus, we have $\int_{\Omega}{dx}=\int_{\Omega}\det F^{\prime}(x)dx.$ That is

[TABLE]

Claim: $\det F^{\prime}(x)\equiv{\rm{1}},~{}\forall x\in{\mathbb{R}^{n}}.$ It’s proof by contradiction. Suppose $\exists~{}{x_{0}}\in\mathbb{R}^{n},\det F^{\prime}(x_{0})\neq{\rm{1}}.$ Without loss of generality, we suppose $\det F^{\prime}({x_{0}})>1,$ denote $C=\det F^{\prime}({x_{0}})-1>0$. Since $F\in{C^{1}}$, $\det F^{\prime}(x)\in C$. $\exists~{}\delta>0$, such that $\det F^{\prime}(x)-1\geqslant\frac{C}{2},$ $\forall x\in U\left({{x_{0}},\delta}\right)$.

Choosing $\Omega=U\left({{x_{0}},\delta}\right)$, thus

[TABLE]

it contradicts.

Thus, we obtain the global injectiveity of $F$ by the Theorem 1.6. Forthermore, the image of $F$ is convex. ∎

Before we give the proof of Theorem 1.8, we need the following proposition.

Proposition 5.1**.**

*Let $F=(f,g):\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a local homeomorphism such that for some $s>0,~{}~{}F|_{\mathbb{R}^{2}\backslash D_{s}}$ . If $F$ satisfies the $W$ condition, then

(1) any half Reeb component of $\mathcal{F}(f)$ or $\mathcal{F}(g)$ is a bounded in $\mathbb{R}^{2}$;

(2) If $F$ extends to a local homemorphism $\widetilde{\overline{F}}=\left({\overline{f},\overline{g}}\right):\mathbb{R}^{2}\to\mathbb{R}^{2}$ , $\mathcal{F}(\overline{f})$ and $\mathcal{F}(\overline{g})$ have no half-Reeb components.*

Proof.

Without loss of generality, we consider the $\mathcal{F}(f)$, by contradiction have an unbounded half Reed component. By the process in Proposition 3.1, we assume that $\mathcal{F}(f)$ has a half Reeb component $\mathcal{A}$ such that $\Pi(\mathcal{A})$ is unbounded interval. Furthermore,

[TABLE]

If $\liminf_{x_{k}\rightarrow+\infty}\Phi^{\prime}(x_{k})P(x_{k})=0.$ There exists a subsequence denoted still $\{x_{k}\}$ with $x_{k}\to+\infty$ such that $\Phi^{\prime}(x_{k})P(x_{k})\to 0.$ That is $\lambda(x_{k})P(x_{k})\to 0.$ Since $F(\mathcal{A})$ is bounded, F\big{(}x_{k},H(x_{k})\big{)} converges to a finite value $T$ on compact set $\overline{\mathcal{F}(\mathcal{A})}$. This contradicts the $W$-condition.

If $\liminf_{x_{k}\rightarrow+\infty}\Phi^{\prime}(x_{k})P(x_{k})\neq 0,$ then $\liminf_{x_{k}\rightarrow+\infty}\Phi^{\prime}(x_{k})P(x_{k})>0$. Thus, there exists $C_{0}>C$ and $l>0$ such that $\Phi^{\prime}(x)P(x)>l,\forall x>C_{0}$. There exists $K>0$ such that, take $C>C_{0}$

[TABLE]

And $\Phi(C)-\Phi(C_{0})<K.$

Then

[TABLE]

This contradiction proves the proposition. ∎

6 The Proof of Theorem 1.8

Proof.

By Proposition 5.1, it’s very easy to know the image of $F$ is convex. This implies that $\mathcal{F}(f)$ has a half Reeb component. It contradicts the Proposition 3.1. Thus, we complete the proof. ∎

7 The Proof of Theorem 1.9

Proof.

By similar methods, we can prove the Theorem 1.9 by half Reeb component and Proposition 5.1. ∎

In finally, we prove the Corollary 1.1.

${\bf The~{}~{}Proof~{}~{}of~{}~{}Corollary~{}~{}\ref{c1}}$: Consider $G:\mathbb{R}^{2}\to\mathbb{R}^{2}$ and $G(z)=F(z)-1,\forall z\in\mathbb{R}^{2}$ . $G(z)$ has no positive eigenvalue because $Spec(G)\subset\{z\in{\mathbb{R}^{2}}:{\mathop{\rm Re}\nolimits}(z)<0\}$. So $G$ is injective by Theorem 1.6. Thus, $F$ has a fixed point.

Remark 7.1**.**

It is very important and meaningful to study the relation between half-Reeb component in higher dimensions and the rate of tending to zero of eigenvalues of $DF$.