Citation: Wei Liu. Global injectivity of differentiable maps via W-condition in R2[J]. AIMS Mathematics, 2021, 6(2): 1624-1633. doi: 10.3934/math.2021097
[1] | Masaaki Mizukami . Remarks on smallness of chemotactic effect for asymptotic stability in a two-species chemotaxis system. AIMS Mathematics, 2016, 1(3): 156-164. doi: 10.3934/Math.2016.3.156 |
[2] | Dong Su, Shilin Yang . Automorphism groups of representation rings of the weak Sweedler Hopf algebras. AIMS Mathematics, 2022, 7(2): 2318-2330. doi: 10.3934/math.2022131 |
[3] | Jiali Yu, Yadong Shang, Huafei Di . Existence and nonexistence of global solutions to the Cauchy problem of thenonlinear hyperbolic equation with damping term. AIMS Mathematics, 2018, 3(2): 322-342. doi: 10.3934/Math.2018.2.322 |
[4] | Ying Zeng, Wenjing Yang . Decay of unique global solution for 3D tropical climate model with partial dissipation. AIMS Mathematics, 2023, 8(12): 30882-30894. doi: 10.3934/math.20231579 |
[5] | Muhammad Nazam, Aftab Hussain, Asim Asiri . On a common fixed point theorem in vector-valued b-metric spaces: Its consequences and application. AIMS Mathematics, 2023, 8(11): 26021-26044. doi: 10.3934/math.20231326 |
[6] | Bo Lu, Angmao Daiqing . Cartan-Eilenberg Gorenstein-injective m-complexes. AIMS Mathematics, 2021, 6(5): 4306-4318. doi: 10.3934/math.2021255 |
[7] | Mourad Berraho . On a problem concerning the ring of Nash germs and the Borel mapping. AIMS Mathematics, 2020, 5(2): 923-929. doi: 10.3934/math.2020063 |
[8] | Danhua He, Liguang Xu . Boundedness analysis of non-autonomous stochastic differential systems with Lévy noise and mixed delays. AIMS Mathematics, 2020, 5(6): 6169-6182. doi: 10.3934/math.2020396 |
[9] | Alexander G. Ramm . When does a double-layer potential equal to a single-layer one?. AIMS Mathematics, 2022, 7(10): 19287-19291. doi: 10.3934/math.20221058 |
[10] | Li Wang, Jun Wang, Daoguo Zhou . Concentration of solutions for double-phase problems with a general nonlinearity. AIMS Mathematics, 2023, 8(6): 13593-13622. doi: 10.3934/math.2023690 |
In 1939, Keller ([19]) stated the following Conjecture:
Conjecture 1.1. (Jacobian conjecture). Let F:kn→kn be a polynomial map, where k is a field of characteristic 0. If the determinant for its Jacobian matrix of the polynomial map is a non-zero constant, i.e., detJF(x)≡C∈k∗,∀x∈kn, then F(x) has a polynomial inverse map.
On the long-standing Jacobian conjecture, it is still open even in the case n=2.
A very important result, for example, if k=Cn, is the following theorem.
Theorem 1.1. ([8]) Let F:Cn→Cn is a polynomial map. If F is injective, then F is bijective. Furthermore the inverse is also a polynomial map.
If k=Rn, then one gets
Conjecture 1.2. (Real Jacobian Conjecture, for short, RJC) If F:Rn→Rn is a polynomial map, detJF(x) is not zero in Rn, then F is a injective map.
It is false and Pinchuk [25] constructs a counterexample to RJC for n=2.
In 2007, Belov-Kanel and Kontsevich [20] proved that Conjecture 1.1 is stably equivant to the Dixmier conjecture. Conjecture 1.1 is also equivalent to the statement: Any ternary Engel algebra in characteristic 0 satisfying a system of Capelli identities is a Yagzhev algebra (see [1], Page 263). Moreover, Conjecture 1.1 is also equivant to some other conjectures, such as the Amazing Image Conjecture [10], a special case of the Vanishing conjecture [31]. There are many results on it, see for example ([2,3,9,15,17,25,28]).
Fernandes et al. [11] study the Conjecture 1.1 by the eigenvalues of the Jacobian matrix JF(x) in R2 and obtain:
Theorem 1.2. ([11]) Let F=(f,g):R2→R2 be a differentiable map. For some ε>0, if
Spec(F)∩[0,ε)=∅, | (1.1) |
where Spec(F) denotes the set of all (complex) eigenvalues of Jacobian matrix JF(x), for all x∈R2, then F is injective.
Theorem 1.2 is deep. If the assumption (1.1) is replaced by 0∉ Spec (F), then the conclusion is false, even for polynomial map F, as the counterexample due to Pinchuck [25]. Pmyth and Xavier [30] proved that there exist n>2 and a non-injective polynomial map F such that Spec(F)∩[0,+∞)=∅.
Theorem 1.2 adds a new result on Markus-Yamabe conjecture [24]. This Conjecture has been solved by Gutierrez [13] and Fessler [12] independently in dimension n=2 in 1993. It is false for n⩾3 even for polynomial vector field, see [7].
Theorem 1.2 also implies that the following conjecture is true in dimension n=2.
Conjecture 1.3. ([5], Conjecture 2.1) Let F:Rn→Rn be a C1 map. Suppose there exists ε>0 such that |λ|⩾ε for all the eigenvalues λ of Jacobian matrix JF(x) and all x∈Rn. Then F is injective.
The essential technique is to use the concept of the half-Reeb component (see Definition 2.1 below) to prove Theorem 1.2.
Theorem 1.2 leads to study the eigenvalue conditions of some maps for injectivity in dimension n=2. In 2007, Gutiérrez and Chau [16] studied the geometrical behavior of differentiable maps and the following ∗-condition on the real eigenvalues of JF in R2 by the half-Reeb component method.
For each θ∈R, we denote the linear rotation Rθ by
Rθ:=(cosθ−sinθsinθcosθ) | (1.2) |
and define the map Fθ=Rθ∘F∘R−θ.
Definition 1.1. ([16], ∗-condition) A differentiable F satisfies the ∗-condition if for each θ∈R, there does not exist a sequence R2∋zk→∞ such that, Fθ(zk)→T∈R2 and JFθ(zk) has a real eigenvalue λk→0.
Theorem 1.3. ([16]) Suppose that F:R2→R2 is a differentiable local homeomorphism.
(i) If F satisfies the ∗-condition, then F is injective and its image is a convex set.
(ii) F is a global homeomorphism of R2 if and only if F satisfies the ∗-condition and its image F(R2) is dense in R2.
Since the ∗-condition is somewhat weaker than the condition (1.1), we can obtain Theorem 1.2 from Theorem 1.3 (ⅰ) by a standard procedure.
For other new cases, the essential difficulty is how to prove that the eigenvalues of JF which may be tending to zero imply F is injective. Rabanal [27] extended the ∗-condition to the following B-condition.
Definition 1.2. ([27], B-condition) The differentiable map F:R2→R2 satisfies the B-condition if for each θ∈R, there does not exist a sequence (xk,yk)∈R2 with xk→+∞ such that Fθ(xk,yk)→T∈R2 and JFθ(xk,yk) has a real eigenvalue λk satisfying λkxk→0.
If one replaced the ∗-condition by the B-conditon, then Theorem 1.3 also holds. Moreover, Rabanal obtained the following theorem.
Theorem 1.4. ([27]) Suppose that the differentiable map F:R2→R2 satisfies the B-condition and detJF(z)≠0,∀z∈R2, then F is a topological embedding.
In fact, Theorem 1.4 improves the main result of Gutiérrez [16], see also [26,29].
In 2014, Braun and Venato-Santos [4] considered the relations between the half-Reeb component and the Palais-Smale condition for global injectivity.
Many references on other aspects of the half Reeb component including higher dimensional situations (see [14,21,22,23,28]).
For example, Gutiérrez and Maquera considered the half-Reeb component for the global injectivity in dimension 3.
Theorem 1.5. ([14]) Let Y=(f,g,h):R3→R3 be a polynomial map such that Spec(Y)∩[0,ε)=∅, for some ε>0. If codim(SY)⩾2, then Y is a bijection.
Recently, W. Liu prove the following theorem by the Minimax method.
Theorem 1.6. ([22]) Let F:Rn→Rn be a C1 map, n⩾2. If for some ε>0,
0∉Spec(F)andSpec(F+FT)⊆(−∞,−ε)or(ε,+∞), |
then F is globally injective.
Let us return to study the eigenvalues of JF approaching to zero by the half-Reeb component method in R2.
In this paper, we first define the W-condition. For the convenience of our statement, let us denote the set
P:={P|R+→R+,Pisnondecreasingand∀M>0,thereexistsalargeconstant N which depends on M and P, such that ∫N21P(x)dx>M}.
Obviously, P contains many functions, such as 1, x, xln(x+1), xln(1+x)ln(1+ln(1+x)) and it doesn't include xα,∀α>1;xlnβ(x+1),∀β>1.
Definition 1.3. (W-condition)
A differentiable map F satisfies the W-condition if for each θ∈R (see (1.2)), there does not exist a sequence (xk,yk)∈R2 with xk→+∞ such that Fθ(xk,yk)→T∈R2 and JFθ(xk,yk) has a real eigenvalue λk satisfying λkP(xk)→0, where P∈P.
Remark 1.1. The W-condition obviously contains the ∗-condition and the B-condition. Let P(x)=xln(x+1)∈P, the W-condition with the function P is weaker than the ∗-condition and the B-condition. It seems can't 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 JF must be in the interval (O(xlnβx)−1,∀β>1, O(xlnx(lnlnxlnlnx)2)−1] by the half-Reeb component method.
Remark 1.2. If xk exchanges yk in definition 1.3, then the W-condition is also vaild.
Remark 1.3. For example, let g(x,y) be a C1 function such that g(x,y)=yxlnx where x⩾2. The map F(x,y)=(e−x,g(x,y)) satisfies detJF=−e−x1xlnx≠0. Then, for {xk}⊆[2,+∞), F(xk,0)=(e−xk,0)→P=(0,0), as xk→+∞. JF(xk,0) has a real eigenvalue
1xklnxk=λk→0. |
However, the limit of the product xklnxk is away from zero.
We use the W-condition and obtain the main result.
Theorem 1.7. Let F:R2→R2 be a differentiable local homeomorphism. If F satisfies the W-condition, then F is injective and F(R2) is convex.
Obviously, Theorem 1.7 implies Theorems 1.3 and 1.4(ⅰ). Moreover, we have:
Theorem 1.8. Let F:R2→R2 be a differentiable Jacobian map. If F satisfies the W-condition, then F is a globally injective, measure-preserving map with convex image.
It improves the main results of Gutiérrez [16], Rabanal [27] and Gutiérrez [18].
Because of the injectivity of map F in Theorem 1.8, we obtain the following fixed point theorem.
Corollary 1.1. If F is as in Theorem 1.8 and Spec(F)⊆{z∈C||z|<1}, then F has at most one fixed point.
Another important property on the Keller map as in corollary 1.1 is the theorem B by Cima et al. [6]. They proved that a global attractor for the discrete dynamical system has a unique fixed point.
By the Inverse Function Theorem, the map F in Theorem 1.7 is locally injective at any point in R2. However, in general, it's not a global injective map. So our goal is to give the sufficient conditions to obtain the global injectivity of F. Here, we also use the W condition as a sufficient condition to obtain the following results.
Theorem 1.9. Let F=(f,g):R2→R2 be a local homeomorphism such that for some s>0,F|R2∖Ds is differentiable. If F satisfies the W-condition, then it is a globally injective and F(R2) 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.
The W condition can be also devoted to studying the differentiable map F:R2∖Ds→R2 whose Spec(F) is disjoint with [0,+∞).
Theorem 1.10. Let F=(f,g):R2∖¯Dσ→R2 be a differential map which satisfies the W-condition. If Spec(F)∩[0,+∞)=∅ or Spec(F)∩(−∞,0]=∅, then there exists s⩾σ such that F|R2∖Ds can be extended to an injective local homeomorphism ˜F=(˜f,˜g):R2→R2.
All these works are related to the Jacobian conjecture which can be reduce to that for any dimension n⩾2, a polynomial map F:Cn→Cn of the form F=x+H, where H is cube-homogeneous and JH is symmetry, is injective if Spec(F)={1} (see [3]).
In order to prove our theorems, we need to use the definition and some propositions of the half-Reeb component.
In this section, we will introduce some preparation work on the eigenvalue conditions of Spec(F).
Let h0(x,y)=xy and we consider the set
B={(x,y)∈[0,2]×[0,2]|0<x+y⩽2}. |
Definition 2.1. (half-Reeb component[13]) Let F be a differentiable map from R2→R2 and detJFp≠0,∀p∈R2, Given h∈{f,g}, we will say that A⊆R2 is a half-Reeb component for F(h) (or simply a hRc for F(h))if there exists a homeomorphism H:B→A which is a topological equivalence between F(h)|A and F(h0)|B and such that:
(1) The segment{(x,y)∈B:x+y=2} is sent by H onto a transversal section for the foliation F(h) in the complement of H(1,1); this section is called the compact edge of A;
(2) Both segments {(x,y)∈B:x=0} and {(x,y)∈B:y=0} are sent by H onto full half-trajectories of F(h). These two semi-trajectories of F(h) are called the noncompact edges of A.
The following Propositions connect the half-Reeb components and injectivity of the map F.
Proposition 2.1. ([11]) Suppose that F=(f,g):R2→R2 is a differentiable map such that 0∉Spec(F). If F is not injective, then both F(f) and F(g) have half-Reeb components.
Proposition 2.2. ([11]) Let F=(f,g):R2→R2 be a non-injective, differentiable map such that 0∉Spec(F). Let A be a hRc of F(f) and let (fθ,gθ)=Rθ∘F∘R−θ, where θ∈R and Rθ is in (1.2). If Π(x,y)=x is bounded, where Π:R2→R is given by Π(x,y)=x, then there is an ε>0 such that, for all θ∈(−ε,0)∪(0,ε); F(fθ) has a hRc Aθ such that Π(Aθ)is an interval of infinite length.
In this section, we will establish the essential fact that the W-condition implies non-existence of half-Reeb component.
Let F=(f,g):R2→R2 be a local homeomorphism of R2. For each θ∈R, we denoted by Rθ the linear rotation (see (1.2)):
(x,y)→(xcosθ−ysinθ,xsinθ+ycosθ), |
and
Fθ:=(fθ,gθ)=Rθ∘F∘R−θ. |
In other words, Fθ represents the linear rotation Rθ in the linear coordinates of R2.
Proposition 3.1. A differentiable local homeomorphism F:R2→R2 which satisfies the 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 (fθ,gθ)=Fθ. From Proposition 2.2, there exists some θ∈R, such that F(Aθ) has a half-Reeb component which Π(A) is unbounded interval, where Π(A) denote orthgonal projection onto the first coordinate in A. Therefore ∃ b and a half-Reeb component A, such that [b,+∞)⊆Π(A). Then, for large enough a>b and any x⩾a, the vertical line Π−1(x) intersects exactly the one trajectory αx∩[x,+∞)=x, i.e. x is maximum of the the trajectory Παx. If x⩾a, the intersection αx∩Π−1(x) is compact subset in A.
Thus, we can define the function H:(a,+∞)→R by
H(x)=sup{y:(x,y)∈Π−1(x)∩αx}. |
As F(fθ) is a foliation, one gets
Φ:(a,+∞)→A byΦ(x)=fθ(x,H(x)). |
We can know that Φ is a bounded, monotone strictly function such that, for a full measure subset M⊆(a,+∞).
Since the image of Φ is contained in fθ(Γ) where Γ is compact edge of hRc A, the function Φ is bounded in (a,+∞). Furthermore, Φ is continuous because F(fθ) is a C0 foliation. Since F(fθ) is transversal to Γ, we have Φ is monotone strictly.
For the measure subset M⊆(a,+∞), such that Φ(x) is differentiable on M and the Jacobian matrix of Fθ(x,y) at (x,H(x)) is
JFθ(x,H(x))=(Φ′(x)0∂xgθ(x,H(x))∂ygθ(x,H(x))). |
Therefore, ∀x∈M, Φ′(x)=∂xfθ(x,H(x)) is a real eigenvalue of JFθ(x,H(x)) and we denote it by λ(x):=Φ′(x).
Since F is local homeomorphism, without loss of generality, we assume Φ is strictly monotone increasing, i.e. Φ′(x)>0,∀x∈M. Let any function P∈P, where
P={P|R+→R+,Pis nondecreasing and∀M>0,there exists large constant N which depends on M and P, such that ∫N21P(x)dx>M}.
Claim:
lim infxk→+∞Φ′(xk)P(xk)>0. |
Because P(x) and Φ′(x) are both positive, we can suppose by contradition that
lim infxk→+∞Φ′(xk)P(xk)=0. There exists a subsequence denoted still by {xk}, such that Φ′(xk)P(xk)→0, as xk→+∞. That is λ(xk)P(xk)→0. Since Fθ(A) is bounded, Fθ(xk,H(xk)) converges to a finite value T on compact set ¯Fθ(A). This contradicts the W-condition.
Therefore, there exist a constant a0(a0>2) and a small ε0>0, such that
Φ′(x)P(x)>ε0,∀x⩾a0. |
Since Φ(x) is bounded, there exists L>0, such that
Φ(x)−Φ(a0)⩽L,∀x⩾a0. |
By the definiton of P, we can choose C large enough, such that
∫Ca01P(x)dx>Lε0. |
Thus,
L⩾Φ(C)−Φ(a0)=∫Ca0Φ′(x)dx⩾∫Ca0ε0P(x)dx>L. |
It's a contradiction.
The proof of Theorem 1.7. By contradiction, we suppose that F is not injective. By Proposition 2.1, we have F has a half-Reeb component. This contradicts Proposition 3.1 that implies F has no half-Reeb component if F satisfies the W-condition. Thus, we complete the proof of Theorem 1.7.
The proof of Theorem 1.8. First, we show that the equivalence of the differential Jacobian map and measure-preserving in any dimension n.
For any nonempty measurable set Ω⊂Rn. Since F:Rn→Rn, we can denote V:={F(x)|x∈Ω}. Let the components of F(x) be vi(i=1,2...n), i.e. F(x1,...xn)=(v1(x1,...xn),...vn(x1...xn)). So dv=detJF(x)dx. Since detJF(x)≡1, we have dv=dx.
Therefore, ∫Vdv=∫Ωdx. It implies F preserves measure.
Inversely, let v=F(x),∀x∈Ω. We still denote V={F(x)|x∈Ω}.
Since F preserves measure, one gets ∫Vdv=∫Ωdx.
Combining it with dv=detJF(x)dx, we obtain ∫Vdv=∫ΩdetJF(x)dx.
Thus, we have ∫Ωdx=∫ΩdetJF(x)dx. That is
∫Ω(1−detJF(x))dx=0,∀Ω⊂Rn. |
Claim: detJF(x)≡1,∀x∈Rn. It's proof by contradiction. Suppose ∃x0∈Rn,detJF(x0)≠1. Without loss of generality, we suppose detJF(x0)>1, denote C=detJF(x0)−1>0. Since F∈C1, detJF(x)∈C. ∃δ>0, such that detJF(x)−1⩾C2, ∀x∈U(x0,δ).
Choosing Ω=U(x0,δ), thus
∫U(x0,δ)(1−detJF(x))dx≤∫U(x0,δ)−C2dx=−C2m(U(x0,δ))<0, |
it contradicts.
Next, we obtain the global injectiveity of F by the Theorem 1.7. Forthermore, the image of F is convex.
Before we show that Theorem 1.9, the following proposition is necessary.
Proposition 5.1. Let F=(f,g):R2→R2 be a local homeomorphism such that for some s>0,F|R2∖Ds. If F satisfies the W condition, then
(1) any half Reeb component of F(f) or F(g) is a bounded in R2;
(2) If F extends to a local homemorphism ˜¯F=(¯f,¯g):R2→R2, F(¯f) and F(¯g) have no half-Reeb components.
Proof. By contradiction, without loss of generality, we consider the F(f) has an unbounded half Reed component. By the process in Proposition 3.1, we assume that F(f) has a half Reeb component A such that Π(A) is unbounded interval. Furthermore,
JF(x,H(x))=(Φ′(x)0∂xg(x,H(x))∂yg(x,H(x))). |
If lim infxk→+∞Φ′(xk)P(xk)=0, where P∈P. There exists a subsequence denoted still {xk} with xk→+∞ such that Φ′(xk)P(xk)→0. That is λ(xk)P(xk)→0. Since F(A) is bounded, F(xk,H(xk)) converges to a finite value T on compact set ¯F(A). This contradicts the W-condition.
If lim infxk→+∞Φ′(xk)P(xk)≠0, then lim infxk→+∞Φ′(xk)P(xk)>0. Thus, there exists C0>0 and l>0 such that Φ′(x)P(x)>l,∀x>C0. For C>C0, there exists K>0, such that
∫CC0lP(x)dx>K. |
Since Φ(C)−Φ(C0)<K, we have
K<∫CC0lP(x)dx≤∫CC0Φ′(x)dx<K. |
It contradicts. We complete the proof of Proposition 5.1.
The proof of Theorem 1.9. By Proposition 5.1, it's very easy to know that the image of F is convex. This implies that F(f) has a half Reeb component. It contradicts the Proposition 3. Thus, we complete the proof.
The proof of Theorem 1.10. By similar procedure, we can prove the Theorem 1.10 by half Reeb component and Proposition 5.1.
In finally, we prove the Corollary 1.1.
The proof of Corollary 1.1. We consider G:R2→R2 and G(z)=F(z)−1,∀z∈R2. Thus, G(z) has no positive eigenvalue because of Spec(G)⊂{z∈R2:Re(z)<0}. By Theorem 1.7, we have G is injective. Therefore, F has a fixed point. We complete the proof of the Corollary 1.1.
Remark 5.1. It's very important and meaningful to study the relations between half-Reeb components in higher dimensions and the rate of tending to zero of eigenvalues of JF.
The author declares no conflict of interest in this paper.
[1] | A. Belov, L. Bokut, L. Rowen, J. T. Yu, Automorphisms in Birational and Affine Geometry, Vol. 79, Springer International Publishing Switaerland, 2014. |
[2] | H. Bass, E. Connell, D. Wright, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 2 (1982), 287-330. |
[3] | M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc., 8 (2005), 2201-2205. |
[4] | F. Braun, J. Venato-Santos, Half-Reeb components, Palais-Smale condition and global injectivity of local diffeomorphisms in R3, Publ. Mat., 58 (2014), 63-79. |
[5] |
M. Chamberland, G. Meisters, A mountain pass to the Jacobian conjecture, Canad. Math. Bull., 41 (1998), 442-451. doi: 10.4153/CMB-1998-058-4
![]() |
[6] |
A. Cima, A. Gasull, F. Manosas, The discrete Markus-Yamabe problem, Nonlinear Anal. Theory Methods Appl., 35 (1999), 343-354. doi: 10.1016/S0362-546X(97)00715-3
![]() |
[7] |
A. Cima, A. van den Essen, A. Gasull, E. Hubbers, F. Manosas, A polynomial counterexample to the Markus-Yamabe conjecture, Adv. Math., 131 (1997), 453-457. doi: 10.1006/aima.1997.1673
![]() |
[8] | S. L. Cynk, K. Rusek, Injective endomorphisms of algebraic and analytic sets, Ann. Polo. Math., 1 (1991), 31-35. |
[9] | A. van den Essen, Polynomial Automorphisms and the Jacobian Conjecture, Berlin: Birkhäuser, 2000. |
[10] | A. van den Essen, The amazing image conjecture, Image, 1 (2010), 1-24. |
[11] |
A. Fernandes, C. Gutiérrez, R. Rabanal, Global asymptotic stability for differentiable vector fields of R2, J. Diff. Equat., 206 (2004), 470-482. doi: 10.1016/j.jde.2004.04.015
![]() |
[12] |
R. Fessler, A proof of the two dimensional Markus-Yamabe stability conjecture and a generalization, Ann. Polon. Math., 62 (1995), 45-74. doi: 10.4064/ap-62-1-45-74
![]() |
[13] |
C. Gutiérrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. Henri Poincaré, 12 (1995), 627-671. doi: 10.1016/S0294-1449(16)30147-0
![]() |
[14] | C. Gutiérrez, C. Maquera, Foliations and polynomial diffeomorphisms of R3, Math. Z., 162 (2009), 613-626. |
[15] |
C. Gutiérrez, B. Pires, R. Rabanal, Asymototic stability at infinity for differentiable vector fields of the plane, J. Diff. Equat., 231 (2006), 165-181. doi: 10.1016/j.jde.2006.07.025
![]() |
[16] |
C. Gutiérrez, N. Van. Chau, A remark on an eigenvalue condition for the global injectivity of differentiable maps of R2, Disc. Contin. Dyna. Syst., 17 (2007), 397-402. doi: 10.3934/dcds.2007.17.397
![]() |
[17] |
C. Gutiérrez, R. Rabanal, Injectivity of differentiable maps R2 → R2 at infinity, Bull. Braz. Math. Soc. New Series, 37 (2006), 217-239. doi: 10.1007/s00574-006-0011-4
![]() |
[18] | C. Gutiérrez, A. Sarmiento, Injectivity of C1 maps R2 → R2 at infinity, Asterisque, 287 (2003), 89-102. |
[19] | O. H. Keller, Ganze gremona-transformation, Monatsh. Math., 47 (1929), 299-306. |
[20] |
A. Belov-Kanel, M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Mosc. Math. J., 7 (2007), 209-218. doi: 10.17323/1609-4514-2007-7-2-209-218
![]() |
[21] | W. Liu, Q. Xu, A minimax principle to the injectivity of the Jacobian conjecture, arXiv.1902.03615, 2019. |
[22] | W. Liu, A minimax method to the Jacobian conjecture, arXiv.2009.05464, 2020. |
[23] |
C. Maquera, J. Venato-Santos, Foliations and global injectivity in Rn, Bull. Braz. Math. Soc. New Series, 44 (2013), 273-284. doi: 10.1007/s00574-013-0013-y
![]() |
[24] | L. Markus, H. Yamabe, Global stability criteria for differential system, Osaka Math. J., 12 (1960), 305-317. |
[25] |
S. Pinchuk, A counterexamle to the strong real Jacobian conjecture, Math. Z., 217 (1994), 1-4. doi: 10.1007/BF02571929
![]() |
[26] |
R. Rabanal, An eigenvalue condition for the injectivity and asymptotic stability at infinity, Qual. Theory Dyn. Syst., 6 (2005), 233-250. doi: 10.1007/BF02972675
![]() |
[27] |
R. Rabanal, On differentiable area-preserving maps of the plane, Bull. Braz. Math. Soc. New Series, 41 (2010), 73-82. doi: 10.1007/s00574-010-0004-1
![]() |
[28] |
P. Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. Math., 146 (1997), 647-691. doi: 10.2307/2952457
![]() |
[29] |
P. Rabier, On the Malgrange condition for complex polynomials of two variables, Manuscr. Math., 109 (2002), 493-509. doi: 10.1007/s00229-002-0322-8
![]() |
[30] |
B. Smyth, F. Xavier, Injectivity of local diffeomorphisms from nearly spectral conditions, J. Diff. Equat., 130 (1996), 406-414. doi: 10.1006/jdeq.1996.0151
![]() |
[31] |
W. Zhao, New proofs for the Abhyankar-Gurjar inversion formula and the equivalence of the Jacobian conjecture and the vanishing conjecture, Proc. Amer. Math. Soc., 139 (2011), 3141-3154. doi: 10.1090/S0002-9939-2011-10744-5
![]() |