Asymptotic stability at infinity for bidimensional Hurwitz vector fields
Asymptotic stability at infinity for bidimensional
Hurwitz vector fields ✩
Roland Rabanal1
Abstract
Let X : U → R2 be a differentiable vector field. Set Spc(X) = {eigenvalues of DX(z) : z ∈ U}.
This X is called Hurwitz if Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Suppose that X is Hurwitz and U ⊂ R2
is the complement of a compact set. Then by adding to X a constant v one obtains that the infinity
is either an attractor or a repellor for X + v. That means: (i) there exists an unbounded sequence
of closed curves, pairwise bounding an annulus the boundary of which is transversal to X + v,
and (ii) there is a neighborhood of infinity with unbounded trajectories, free of singularities and
periodic trajectories of X+v. This result is obtained after to proving the existence of X̃ : R2 → R2,
a topological embedding such that X̃ equals X in the complement of some compact subset of U.
Keywords: Injectivity, Reeb Component, Asymptotic Stability
2008 MSC: Primary: 37E35, 37C10; Secondary: 26B10, 58C25
1. Introducion1
A basic example of non–discrete dynamics on the Euclidean space is given by a linear vector2
field. This linear system is infinitesimally hyperbolic if every eigenvalue has nonzero real part,3
and it has well known properties [16, 3]. For instance, when the real part of its eigenvalues are4
negative (Hurwitz matrix), the origin is a global attractor rest point. In the nonlinear case, there5
has been a great interest in the local study of vector fields around their rest points [6, 27, 7, 28].6
However, in order to describe a global phase-portrait, as in [22, 25, 5, 8, 9] it is absolutely7
necessary to study its behavior in a neighborhood of infinity [19].8
The Asymptotic Stability at Infinity has been investigated with a strong influence of [18],9
where Olech showed a connection between stability and injectivity (see also [10, 4, 11, 26, 22]).10
This research was also studied in [19, 12, 14, 15, 23, 1]. In [12], Gutierrez and Teixeira study11
C1−vector fields Y : R2 → R2, the linearizations of which satisfy (i) det(DY(z)) > 0 and12
(ii) Trace(DY(z)) < 0 in an neighborhood of infinity. By using [9], they prove that if Y has a13
rest point and the Index I(Y) =
Trace (DY) < 0 (resp. I(Y) ≥ 0), then Y is topologically14
equivalent to z 7→ −z that is “the infinity is a repellor ”(resp. to z 7→ z that is “the infinity15
is an attractor”). This Gutierrez-Teixeira’s paper was used to obtain the next theorem, where16
Spc(Y) = {eigenvalues of DY(z) : z ∈ R2 \ Dσ}, andℜ(z) is the real part of z ∈ C.17
✩This paper was written when the author served as an Associate Fellow at ictp-Italy.
Email address: rrabanal@pucp.edu.pe (Roland Rabanal)
1The author was partially supported by pucp-Peru (dai: 2012-0020).
Preprint submitted to Elsevier November 16, 2018
http://arxiv.org/abs/0704.1418v4
Theorem 1 (Gutierrez-Sarmiento). Let Y : R2 \ Dσ → R2 be a C1− map, where σ > 0 and18
Dσ = {z ∈ R
2 : ||z|| ≤ σ}. The following is satisfied:19
(i) If for some ε > 0, Spc(Y) ∩ (−ε,+∞) = ∅. Then there exists s ≥ σ such that the restriction20
Y | : R2 \ Ds → R
2 is injective.21
(ii) If for some ε > 0, the spectrum Spc(Y) is disjoint of the union (−ε, 0]∪{z ∈ C : ℜ(z) ≥ 0}.22
Then there exist p0 ∈ R
2 such that the point ∞ of the Riemann Sphere R2 ∪ {∞} is either23
an attractor or a repellor of z′ = Y(z) + p0.24
Theorem 1 is given in [14], and it has been extended to differentiable maps X : R2 \Dσ → R
in [13, 15]. In both papers the eigenvalues also avoid a real open neighborhood of zero. In [23]26
the author examine the intrinsic relation between the asymptotic behavior of Spc(X) and the27
global injectivity of the local diffeomorphism given by X. He uses Yθ = Rθ ◦Y ◦R−θ, where Rθ is28
the linear rotation of angle θ ∈ R, and (motivated by [11]) introduces the so–called B−condition29
[24, 26], which claims:30
for each θ ∈ R, there does not exist a sequence (xk, yk) ∈ R
2 with xk → +∞ such that31
Yθ((xk, yk))→ p ∈ R
2 and DYθ(xk, yk) has a real eigenvalue λk satisfying xkλk → 0.32
By using this, [23] improves the differentiable version of Theorem 1.33
In the present paper we prove that the condition34
Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}
is enough in order to obtain Theorem 1 for differentiable vector fields X : R2 \ Dσ → R
Throughout this paper, R2 is embedded in the Riemann sphere R2∪{∞}. Thus (R2\Dσ)∪{∞}36
is the subspace of R2 ∪ {∞} with the induced topology, and ‘infinity ’refers to the point ∞ of37
2 ∪ {∞}. Moreover given C ⊂ R2, a closed (compact, no boundary) curve (1−manifold),38
D(C) (respectively D(C)) is the compact disc (respectively open disc) bounded by C. Thus, the39
boundaries ∂D(C) and ∂D(C) are equal to C, homeomorphic to ∂D1 = {z ∈ R
2 : ||z|| = 1}.40
2. Statements of the results41
For every σ > 0 let Dσ = {z ∈ R
2 : ||z|| ≤ σ}. Outside this compact disk we consider a42
differentiable vector field X : R2 \Dσ → R
2. As usual, a trajectory of X starting at q ∈ R2 \Dσ43
is defined as the integral curve determined by a maximal solution of the initial value problem44
ż = X(z), z(0) = q. This is a curve Iq ∋ t 7→ γq(t) = (x(t), y(t)), satisfying:45
• t varies on some open real interval containing the zero, the image of which γq(0) = q;46
• γq(t) ∈ R
2 \ Dσ and there exist the real derivatives
(t), dy
(t);47
• γ̇q(t) =
(t), dy
the velocity vector field of γq at γq(t) equals X(γq(t)) and48
• Iq ⊂ R is the maximal interval of definition.49
We identify the trajectory γq with its image γq(Iq), and we denote by γ
q (resp. γ
q ) the positive50
(resp. negative) semi-trajectory of X, contained in γq and starting at q. In this way γq = γ
q ∪ γ
q .51
Thus each trajectory has its two limit sets, α(γ−q ) and ω(γ
q ) respectively. These limit sets are52
well defined in the sense that they only depend on the respective solution.53
A C0−vector field X : R2 \ Dσ → R
2 \ {0} (without rest points) can be extended to a map54
X̂ : ((R2 \ Dσ ∪∞),∞) −→ (R
2, 0)
(which takes∞ to 0) [1]. In this manner, all questions concerning the local theory of isolated rest55
points of X can be formulated and examined in the case of the vector field X̂. For instance, if γ+p56
(resp. γ−p ) is an unbounded semi-trajectory of X : R
2 \ Dσ → R
2 starting at p ∈ R2 \ Dσ with57
empty ω−limite (resp. α−limit) set, we say γ+p goes to infinity (resp. γ
p comes from infinity),58
and it is denoted by ω(γ+p ) = ∞ (resp. α(γ
p ) = ∞). Therefore, we may also talk about the phase59
portrait of X in a neighborhood of∞.60
As in our paper [15], we say that the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is61
an attractor (resp. a repellor) for the continuos vector field X : R2 \ Dσ → R2 if:62
• There is a sequence of closed curves, transversal to X and tending to infinity. That is63
for every r ≥ σ there exist a closed curve Cr such that D(Cr) contains Dr and Cr has64
transversal contact to each small local integral curve of X at any p ∈ Cr .65
• For some Cs with s ≥ σ, all trajectories γp starting at a point p ∈ R
2 \ D(Cs) satisfy66
ω(γ+p ) = ∞ that is γ
p go to infinity (resp. α(γ
p ) = ∞ that is γ
p come from infinity).67
We also recall that I(X), the index of X at infinity is the number of the extended line [−∞,+∞]68
given by69
I(X) =
Trace(DX̂)dx ∧ dy,
where X̂ : R2 → R2 is a global differentiable vector field such that:70
• In the complement of some disk Ds with s ≥ σ both X and X̂ coincide.71
• The map z 7→ Trace(DX̂z) is Lebesgue almost–integrable in whole R
2, in the sense of [15].72
This index is a well-defined number in [−∞,+∞], and it does not depend on the pair (X̂, s) as73
shown [15, Lemma 12].74
Definition 1. The differentiable vector field (or map) X : R2 \ Dσ → R2 is called Hurwitz if75
every eigenvalue of the Jacobian matrices has negative real part. This means that its spectrum76
Spc(X) = {eigenvalues of DX(z) : z ∈ R2 \ Dσ} satisfies Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}.77
We are now ready to state our result.78
Theorem 2. Let X : R2 \ Dσ → R2 be a differentiable vector field (or map), where σ > 0 and79
Dσ = {z ∈ R
2 : ||z|| ≤ σ}. Suppose that X is Hurwitz: Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Then80
(i) There are s ≥ σ and a globally injective local homeomorphism X̃ : R2 → R2 such that X̃81
and X coincide on R2 \ Ds. Moreover, the restriction X| : R
2 \ Ds → R
2 is injective, and82
it admits a global differentiable extension X̂ such that the pair (X̂, s) satisfies the definition83
of the index of X at infinity, and this index I(X) ∈ [−∞,+∞).84
(ii) For all p ∈ R2 \Dσ, there is a unique positive semi-trajectory of X starting at p. Moreover,85
for some v ∈ R2, the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is an attractor86
(respectively a repellor) for the vector field X+v : R2\Ds → R
2 as long as the well-defined87
index I(X) ≥ 0 (respectively I(X) < 0).88
The map X̃ of Theorem 2 is not necessarily a homeomorphism. This X̃ is a topological89
embedding, the image of which may be properly contained in R2. Furthermore, if A : R2 → R290
is an arbitrary invertible linear map, then Theorem 2 applies to the map A ◦ X ◦ A−1.91
Theorem 2 improves the main results of [13, 15]. Item (i) complements the injectivity work92
of [13] (see also [23, 14]), where the authors consider the assumption Spc(X) ∩ (−ε,+∞) = ∅93
(as in Theorem 1). Item (ii) generalizes [15], where the authors utilize the second condition of94
Theorem 1. In our new assumptions, the negative eigenvalues can tend to zero.95
2.1. Description of the proof of Theorem 296
Since the Local Inverse Function Theorem is true, a map X = ( f , g) : R2 \ Dσ → R
as in Theorem 2 is a local diffeomorphism. Thus the level curves { f = constant} make up a98
C0−foliation F ( f ) the leaves of which are differentiable curves, and the restriction of the other99
submersion g to each of these leaves is strictly monotone. In particular, F ( f ) and F (g) are100
(topologically) transversal to each other. We orient F ( f ) in agreement that if Lp( f ) is an oriented101
leaf of F ( f ) thought the point p, then the restriction g| : Lp( f ) → R is an increasing function102
in conformity with the orientation of Lp( f ). We denote by L
p ( f ) = {z ∈ Lp( f ) : g(z) ≥ g(p)}103
(resp. L+p(g) = {z ∈ Lp(g) : f (z) ≥ f (p)}) and L
p( f ) = {z ∈ Lp( f ) : g(z) ≤ g(p)} (resp.104
L−p (g) = {z ∈ Lp(g) : f (z) ≤ f (p)}) the respective positive and negative half-leaf of F ( f ) (resp.105
F (g)). Thus Lq( f ) = L
q ( f ) ∪ L
q ( f ) and L
q ( f ) ∩ L
q ( f ) = {q}. In this context, the nonsingular106
vector fields107
X f = (− fy, fx) and X̃g = (gy,−gx), (2.1)
given by the partial derivatives are tangent to Lp( f ) and Lp(g), respectively. This construction108
has previously been used in [4].109
We need the following definition [14, 4]. Let h0(x, y) = xy and consider the set110
(x, y) ∈ [0, 2] × [0, 2] : 0 < x + y ≤ 2
Definition 2. Let X = ( f , g) be a differentiable local homeomorphism. Given h ∈ { f , g}, we say112
that A (in the domain of X) is a half-Reeb component for F (h) if there is a homeomorphism113
H : B→ A which is a topological equivalence between F (h)|A and F (h0)|B such that: (Fig. 3)114
• The segment {(x, y) ∈ B : x+ y = 2} is sent by H onto a transversal section for the foliation115
F (h) in the complement of the point H(1, 1); this section is called the compact edge ofA.116
• Both segments {(x, y) ∈ B : x = 0} and {(x, y) ∈ B : y = 0} are sent by H onto full117
half-leaves of F (h). These half-leaves of F (h) are called the non–compact edges ofA.118
Observe thatAmay not be a closed subset of R2, and H does not need to be extended to infinity.119
Section 3 gives new results on the foliations induced by a local diffeomorphism X = ( f , g) :120
2 \ Dσ → R
2 (see also Proposition 1). Theorem 3 implies that the conditions121
Spc(X) ∩ [0,+∞) = ∅ and ‘each half–Reeb component of F ( f ) is bounded ’ (2.2)
give the existence of s ≥ σ such that X|
R2\Ds
can be extended to an injective map X̃ : R2 → R2.122
Section 4 presents some preliminary results on maps such that Spc(X) ∩ [0,+∞) = ∅. Section 5123
concludes with the proof of Theorem 2. The main step is given in Proposition 2, which implies124
that Hurwitz maps satisfy (2.2). Therefore, Theorem 2 is obtained by using this Proposition 2125
and some previous work [13, 15].126
3. Local diffeomorphisms that are injective on unbounded open sets127
Let X = ( f , g) : R2 \ Dσ → R
2 be an orientation preserving local diffeomorphism, that is128
det(DX) > 0. Next subsection gives preparatory results about F ( f ) in order to obtain that X will129
be injective on topological half planes (see Proposition 1). Subsection 3.3 presents a condition130
under which X is injective at infinity, that is outside some compact set.131
3.1. Avoiding tangent points132
Let C ⊂ R2 \ Dσ be a closed curve surrounding the origin. We say that the vector field133
X : R2 \ Dσ → R
2 has contact (resp. tangency with; resp. transversal to; etc) with C at p ∈ C if134
for each small local integral curve of X at p has such property.135
Definition 3. A closed curve C ⊂ R2 \ Dσ is in general–position with F ( f ) if there exists a set136
T ⊂ C, at most finite such that:137
• F ( f ) is transversal to C \ T .138
• F ( f ) has a tangency with C at every point of T .139
• A leaf of F ( f ) can meet tangentially C at most at one point.140
Denote by GP( f , s) the set of all closed curves C ⊂ R2 \ Dσ in general–position with F ( f )141
such that Ds ⊂ D(C). If C ⊂ R
2 \ Dσ is in general–position, we denote by n
e(C, f ) (resp.142
ni(C, f )) the number of tangent points of F ( f ) with C, which are external (resp. internal). Here,143
external (resp. internal) means the existence of a small open interval ( p̃, q̃) f ⊂ Lp( f ) such that144
the intersection set ( p̃, q̃) f ∩C = {tangent point} and ( p̃, q̃) f ⊂ R
2 \ D(C) (resp. ( p̃, q̃) f ⊂ D(C)).145
Remark 1. If C ⊂ R2 \ Dσ is in general–position with F ( f ), there exist a, b ∈ C two different146
point such that f (C) = [ f (a), f (b)]. Moreover, a and b are external tangent points because the147
map ( f , g) is an orientation preserving local diffeomorphism. Since C and f (C) are connected and148
C is not contained in any leaf of F ( f ), we conclude that both external tangencies are different.149
Corollary 1 gives important properties of the leafs passing trough a point in Remark 1, if we150
select C ∈ GP( f , σ) with the minimal number of internal tangencies. To this end, the next lemma151
will be needed.152
Lemma 1. Let C ∈ GP( f , σ). Suppose that a leaf Lq( f ) of F ( f ) meets C transversally some-153
where and with an external tangency at a point p ∈ C. Then Lq( f ) contains a closed subinterval154
[p, r] f which meets C exactly at {p, r} (doing it transversally at r) and the following is satisfied:155
(a) If [p, r] is the closed subinterval of C such that Γ = [p, r] ∪ [p, r] f bounds a compact disc156
D(Γ) contained in R2 \D(C), then points of Lq( f ) \ [p, r] f nearby p do not belong to D(Γ).157
(b) Let ( p̃, r̃) and [ p̃, r̃] be subintervals of C satisfying [p, r] ⊂ ( p̃, r̃) ⊂ [ p̃, r̃]. If p̃ and r̃ are158
close enough to p and r, respectively; then we may deform C into C1 ∈ GP( f , σ) in such159
a way that the deformation fixes C \ ( p̃, r̃) and takes [ p̃, r̃] ⊂ C to a closed subinterval160
[ p̃, r̃]1 ⊂ C1 which is close to [p, r] f . Furthermore, the number of generic tangencies of161
F ( f ) with C1 is smaller than that of F ( f ) with C.162
Proof. We refer the reader to [14, Lemma 2].163
Corollary 1. Let C ∈ GP( f , s). Suppose that C minimizes ni(C, f ) and f (C) = [ f (a), f (b)], then164
C ∩ La( f ) = {a} and C ∩ Lb( f ) = {b}.165
Proof. We only consider the case of the point a. Assume by contradiction that the number of166
elements in C ∩ La( f ) is greater than one i.e ♯(C ∩ La( f )) ≥ 2. The last condition of Definition 3167
implies that the intersection of La( f ) with the other point is transversal to C. Then there is a disk168
as in statement (a) of Lemma 1. By using the second part of Lemma 1 we can avoid the external169
tangency a and some internal tangency. This is a contradiction because ni(C, f ) is minimal.170
Remark 2. Corollary 1 remains true if we take any external tangency (not necessarily a and b)171
in a closed curve Cs ∈ GP( f , s) with minimal n
i(Cs, f ).172
3.2. Minimal number of internal tangent points173
A oriented leaf of F ( f ) whose distance to Dσ is different from zero has unbounded half–174
leaves. Given any Lp( f ) with unbounded half–leaves, we denote by H
+(Lp) and H
−(Lp) the two175
components of R2 \ Lp( f ) in order that L
p(g) \ {p} be contained in H
+(Lp). Therefore, the image176
X(H+(Lp)) is an open connected subset of the semi–plane {x > f (p)} := {(x, y) ∈ R
2 : x > f (p)}.177
Remark 3. If the image X(H+(Lp)) is a vertical convex set, all the level curves { f = c} ⊂ H+(Lp)178
are connected. Thus the restriction X| : H+(Lp) → X(H
+(Lp)) is an homeomorphism, and it179
sends every leaf of F ( f )|H+(Lp) over vertical lines. Therefore, it is a topological equivalence180
between two foliations.181
Lemma 2. If H+(Lp) is disjoint from Dσ and the image X(H+(Lp)) is not a vertical convex set,182
then H+(Lp) contains a half-Reeb component of F ( f ).183
Proof. By remark 3, some level set { f = c̃} ⊂ H+(Lp) is disconnected. Therefore, the result is184
obtained directly from [11, Proposition 1].185
Lemma 2 and Remark 3 hold when we consider H−(Lp).186
Lemma 3. Recall that GP( f , s) is the set of all closed curves C ⊂ R2 \ Dσ in general–position187
with F ( f ) such that Ds ⊂ D(C). Let η
i : [σ,∞) → N ∪ {0} be the function given by ηi(s) =188
ni(Cs, f ) where Cs ∈ GP( f , s) minimizes the number of internal tangent points with F ( f ). The189
following statements hold:190
(a) The function ηi is nondecreasing.191
(b) If ηi is bounded then, there exist s0 ∈ [σ,∞) such that η
i(s) ≤ ηi(s0) for all s ∈ [σ,∞).192
(c) Set f (Cs0 ) = [ f (a), f (b)]. Suppose that F ( f ) has a half-Reeb componentA whose image193
f (A) is disjoint from ( f (a), f (b)), then such s0 ∈ [σ,∞) is not a maximum value of the194
function ηi.195
Proof. As Cs+1 also belongs to GP( f , s) we have that η
i(s) ≤ ni(Cs+1, f ). Therefore (a) is true.196
To prove the second part, we introduce the set As = {n ∈ N ∪ {0} : n ≥ n
i(Cs, f ) = η
i(s)},197
for every s ≥ σ. From this definition it is not difficult to check that: ηi is bounded if and only if198
∩s≥σAs , ∅. Therefore, the first element of ∩s≥σAs , ∅ is the bound η
i(s0) of statement (b). This199
proves the second statement.200
We shall have established (c) if we prove that there is some Cs0+ε with ε > 0 such that201
ηi(s0) + 1 ≤ n
i(Cs0+ε, f ). To this end, we select s > s0 large enough for which Cs is enclosing202
D(Cs0 )∪Γwhere Γ is the compact edge ofA. Since, this Cs intersects both leaves Lp( f ) and Lq( f )203
where p and q are the endpoints of Γ, we obtain that ni(Cs, f ) is greater than n
i(Cs0 , f ) = η
i(s0).204
Therefore, for some ε > 0 there is Cs0+ε such that η
i(s0) + 1 ≤ n
i(Cs0+ε, f ). This proves (c).205
Proposition 1. Let X = ( f , g) : R2 \Dσ → R2 be a map with det(DX) > 0. Consider Cs and ηi as206
in Lemma 3. If Cs0 satisfies that n
i(Cs, f ) ≤ n
i(Cs0 , f ) for all s ∈ [σ,∞) and f (Cs0 ) = [ f (a), f (b)].207
Then, for each p ∈ {a, b} at least one of the restrictions of X to H+(Lp) or H
−(Lp) is a globally208
injective map, in agrement that the domain of this restriction is in the complement of D(Cs0 ).209
Proof. We only consider the case p = a. Suppose that H+(La) is contained in the complement210
of D(Cs0). From Remark 3 it is sufficient to prove that X(H
+(La)) is vertical convex. Suppose211
by contradiction that it is false. Then Lemma 2 implies that there is a half-Reeb component212
A ⊂ H+(La). By using statement (c) of Lemma 3 this s0 is not a maximum value of the function213
ηi. This contradiction with our selection of the circle Cs0 conclude the proof.214
3.3. Extending maps to topological embeddings215
The next theorem implies the injectivity at infinity of a map, and it is obtained by using216
the methods, ideas and arguments of [13]. We only give the proof, in the case of continuously217
differentiable maps.218
Theorem 3. Let X = ( f , g) : R2 \ Dσ → R2 be an differentiable local homeomorphism with219
det(DX) > 0. Suppose that Spc(X)∩ [0,+∞) = ∅, and each half–Reeb component of either F ( f )220
or F (g) is bounded. Then there exist s ≥ σ such that the restriction X| : R2 \ Ds → R
2 can be221
extended to a globally injective local homeomorphism X̃ = ( f̃ , g̃) : R2 → R2.222
Proof. We can apply the results of [16, pp. 166-174] to the continuous vector field X f and obtain223
that for each closed curve C ∈ GP( f , σ) the Index of X f along C, denoted by Ind(X f ; C), satisfies224
Ind(X f ; C) =
2 − ne( f ,C) + ni( f ,C)
If X f is discontinuous, we proceed as in [13] by using the index of the foliation F ( f ) which also225
satisfies this formulae.226
(a.1) We claim that ne( f ,C) = ni( f ,C) + 2, for all C ∈ GP( f , σ).227
Suppose that (a.1) is false, so there is C̃1 ∈ GP( f , σ) whose Ind(X f ; C̃
1) , 0. Thus, for228
some point in C̃1 the Hamiltonian X f (p) = (− fy(p), fx(p)) is vertical. More precisely we can229
obtain p ∈ C̃1 such that fy(p) = 0 and fx(p) > 0. This is a contradiction with the eigenvalue230
assumptions because fx(p) ∈ Spc(X) ∩ (0,+∞). (If X f is discontinuous, we refer the reader to231
[13, Proposition 3.1] where proves that the index of the foliation F ( f ) is zero). Therefore, (a.1)232
holds.233
(a.2) We claim that if Cσ ⊂ R
2 \ Ds minimizes n
i( f ,Cσ), then every internal tangency in Cσ234
gives a half–Reeb component.235
For every internal tangency q ∈ Cσ we consider the forward Poincaré map T : [p, q)σ ⊂236
Cσ → Cσ induced by the oriended F ( f ) (if T : (q, r]σ ⊂ Cσ → Cσ the proof is similar) where237
[p, q)σ ⊂ Cσ is the maximal connected domain of definition of T on which this first return map238
is continuous. If the open arc L+p ( f ) \ {p} intersects Cσ we apply Lemma 1, so we can deform239
Cσ in a new circle C̃
1 ⊂ R2 \ Ds such that the number of internal tangencies of C̃
1 with F ( f )240
is (strictly) smaller than that of Cσ. This is a contradiction. Therefore L
p( f ) \ {p} is disjoint241
from Cσ. By using this and our selection of [p, q)σ ⊂ Cσ is not difficult to check that there is a242
half–Reeb component of F ( f ) whose compact edge is contained in Cs, Thus, we obtain (a.2).243
Notice that, for every circle as in (a.2) any internal tangency of this Cσ gives an unbounded244
half–Reeb component, thus by our assumptions and (a.1) we have that ni( f ,Cσ) = 0. Therefore,245
(a.3) if Cσ ∈ GP( f , σ) is as in (a.2) then n
i( f ,Cσ) = 0 and n
e( f ,Cσ) = 2. Moreover, F ( f ),246
restricted to R2 \ D(Cσ), is topologically equivalent to the foliation made up by all the247
vertical straight lines, on R2 \ D1.248
Since X has no unbounded half–Reeb component, we can use the last section of [13] (see249
Proposition 5.1) and obtain that the closed curve Cσ of (a.3) can be deformed so that, the resulting250
new circle C has an exterior collar neighborhood U ⊂ R2 \ D(C) such that:251
(b) X(C) is a non-trivial closed curve, X(U) is an exterior collar neighborhood of X(C) and the252
restriction X| : U → X(U) is a homeomorphism.253
By Schoenflies Theorem [2] the map X| : C → X(C) can be extended to a homeomorphism254
X1 : D(C) → D(X(C)). We extend X : R
2 \ D(C) → R2 to X̃ = ( f̃ , g̃) : R2 → R2 by defining255
X̃|D(C) = X1. Thus X̃| : U → X(U) is a homeomorphism and U (resp. X(U)) is a exterior256
collar neighborhood of C (resp. X(C)). Consequently, X̃ is a local homeomorphism and F ( f̃ ) is257
topologically equivalent to the foliation made up by all the vertical straight lines. The injectivity258
of X̃ follows from the fact that F ( f̃ ) in trivial [4, Proposition 1.4]. This concludes the proof.259
Corollary 2. Suppose that X satisfies Theorem 3. Then the respective extension X̃ = ( f̃ , g̃) :260
2 → R2 of X is a globally injective local homeomorphism the foliations of which, F ( f̃ ) and261
F (g̃) have no half-Reeb components.262
Proof. We reefer the reader to affirmation (a.3) in the proof of Theorem 3.263
Corollary 3. Suppose that X = ( f , g) : R2 \ Dσ → R2 is an orientation preserving local264
diffeomorphism. Then the foliation F ( f ) (resp. F (g)) has at most countably many half–Reeb265
components.266
Proof. A half–Reeb component has a tangency with some Cn of Lemma 3 with n ∈ N large267
enough. We conclude, since the closed curves has at most a finite number of tangent points.268
Remark 4. By using a smooth embedding ( f , g) : R2 → R2, the authors of [10, Proposition 1]269
prove the existence of foliations F ( f ) which have infinitely many half-Reeb components.270
4. Maps free of positive eigenvalues271
In this section we present some properties of a map the spectrum of which is disjoint of272
[0,+∞). These results will be used in Section 5 to proving the first part of Theorem 2. In this273
context, we consider F ( f ) and their trajectories Lq = Lq( f ), L
q = L
q ( f ) and L
q = L
q ( f ).274
Lemma 4. Let X = ( f , g) : R2 \ Dσ → R2 be a differentiable local homeomorphism. Suppose275
that the spectrum Spc(X) ∩ [0,+∞) = ∅ and p = (a, c). Then the intersection of L+p with the276
vertical ray {(a, y) ∈ R2 : y ≥ c} is the one point set {p}.277
Proof. Assume, by contradiction, that L+p \ {p} intersects {(a, y) ∈ R
2 : y ≥ c}. We take d > c278
the smallest value such that q = (a, d) ∈ L+p (see Figure 1a). We only consider the case in which279
the compact arc [p, q] f ⊂ L
p such that Π([p, q] f ) equals the interval [a, a0] with a < a0, where280
Π(x, y) = x (in the other case, Π([p, q] f ) = [a0, a], a0 < a the argument is similar). Therefore, if281
we take the vertical segment [p, q]a = {(a, y) : c ≤ y ≤ d} joint to the open disk D(C) bounded282
by the closed curve C = [p, q] f ∪ [p, q]a. We meet two possible cases:283
PSfrag replacements
(a, c)(a, c)
(a, d)(a, d)
(a0, c0)(a0, c0)
Figure a Figure b
Figure 1:
The first one is that D(C)∩Dσ = ∅. We select the point (a0, c0) ∈ [p, q] f in order that c0 will284
be the smallest value of the compact set Π−1(a0) ∩ [p, q] f . Let R be the closed region bounded285
by the union of {(a, y) : y ≤ c}, {(a0, y) : y ≤ c0} and [p, q0] f ⊂ [p, q] f where q0 = (a0, c0). As286
c0 = inf{y ∈ Π
−1(a0) : (a0, y) ∈ [p, q] f } the compact arc [p, q] f is tangent to the vertical line287
−1(a0) at the point (a0, c0). Thus, X f (a0, c0) is vertical, and so fy(a0, c0) = 0. This implies that288
fx(a0, c0) ∈ Spc(X). By the assumptions about Spc(X), fx(a0, c0) < 0 which in turn implies that289
the arc [q0, q] f ⊂ [p, q] f must enter into R and cannot cross the boundary of R (see Figure 1b).290
This contradicts the fact that q = (a, d) < R.291
The second case happens when D(C) ∩ Dσ , ∅. As [p, q] f is not contained in Dσ, either292
σ < a0 or σ = a0. If σ < a0, the vertical line x = σ meet [p, q] f in two different points293
which define a closed curve as in Figure 1a such that it bounds an open disk disjoint of Dσ. We294
conclude by using the proof of the first case. If σ = a0, we observe the continuous foliation295
F ( f ) in a neighborhood of [p, q] f and meet two points p̃ = (σ, c̃) and q̃ = (σ, d̃) with c̃ < d̃,296
but Π([ p̃, q̃] f ) ⊂ [σ,+∞). It satisfies the conditions of the first case. Therefore the lemma is297
proved.298
Remark 5. Lemma 4 remains true, if we consider the negative leaf L−q starting at q = (a, d) joint299
to the vertical ray {(a, y) ∈ R2 : y ≤ d}.300
Lemma 5. Let X = ( f , g) : R2 \ Dσ → R2 be a map with Spc(X) ∩ [0,+∞) = ∅. Consider301
L+p ⊂ { f = f (p)} and the projection Π(x, y) = x. If the oriented compact arc [p, q] f ⊂ L
p and its302
image Π([p, q] f ) is the interval [Π(p),Π(q)] ⊂ (σ,+∞) with Π(p) < Π(q). Then303
[p,q] f
〈X,∇ f 〉dt ≥ f (p)
[p,q] f
fx dt + g(p)
Π(q) − Π(p)
where 〈 , 〉 denotes the usual inner product on the plane, and fx is the first partial derivative.304
Proof. For each α ∈ Π([p, q] f ) = [a, b], the vertical line Π
−1(α) intersects [p, q] f in a non-empty305
compact set. So there exist ŷα = sup{y ∈ R : (y, α) ∈ [p, q] f } and mα = (α, ŷα). We also define306
S ⊂ (a, b) as the set of critical values of the projection Π(x, y) = x restricted to the differentiable307
PSfrag replacements
aa bb
(α, c)
[Π(p),Π(q)]c
Figure a Figure b
Figure 2: Here
∇ f (z), X f (z)
a positive basis
arc [p, q] f . By the Sard’s Theorem, presented in [17, Theorem 3.3] this set S is closed and has308
zero Lebesgue measure. For α ∈ (a, b)\S the setΠ−1(α) intersects [p, q] f in at most finitely many309
points and the complement set
−1(α)∩[p, q] f
\{mα} is either empty or its cardinality is odd. By310
Lemma 4, the order of these points in the line Π−1(α) oriented oppositely to the y−axis coincides311
with that on the oriented arc [p, q] f (a behavior as in Figure 1a does not exist). Therefore, for312
every α ∈ (a, b) \ S the set
−1(α) ∩ [p, q] f
\ {mα} splits into pairs {pα = (α, dα), qα = (α, cα)}313
with the following three properties: (a.1) cα < dα, (a.2) the compact arc [pα, qα] f lies in the314
semi-plane {x ≤ α} and it is oriented from pα to qα (a.3) g(qα) > g(pα). Notice that the tangent315
vector of [p, q] f at pα has a negative x−component: fy(pα) > 0, and the respective tangent vector316
at qα satisfies fy(qα) < 0. Similarly, fy(mα) ≤ 0 (see (2.1)).317
Assertion Take c < inf{y : (α, y) ∈ [p, q] f } and [Π(p),Π(q)]c = {(α, c) : a ≤ α ≤ b}, an318
horizontal segment. Consider the compact set D(C) ⊂ {(x, y) : a ≤ x ≤ b} the boundary319
of which contain [p, q] f ∪ [Π(p),Π(q)]c (see Figure 2b). Suppose that C, the boundary of320
D(C) is negatively oriented (clock wise). Then321
gy(x, y)dx ∧ dy =
[p,q] f
〈F,∇ f 〉dt −
g(α, c)dα,
where F(x, y) =
f (x, y) − f (p), g(x, y)
Proof of Assertion We will use the Green’s formulae given in [21, Corollary 5.7] (see also [20])323
with the differentiable map G : z 7→ (0, g(z)) and the outer normal vector of C denoted by324
z 7→ η(z) (unitary). By using that Trace(DGz) = gy(z) it follows that325
gy(x, y)dx ∧ dy =
〈G,−η〉ds, (4.1)
where ds denotes the arc length element. If [Π(q),Π(p)]c denote [Π(p),Π(q)]c oriented from326
Π(q) to Π(p), then C = [p, q] f ∪ B ∪ [Π(q),Π(p)]c ∪ A with A and B two oriented vertical327
segments. Consequently,328
〈G,−η〉ds =
[p,q] f
〈G,−η〉ds +
〈G,−η〉ds +
[Π(q),Π(p)]c
〈G,−η〉ds +
〈G,−η〉ds.
In A∪B the vector −η is horizontal i.e η(z) = (η1(z), 0) then G = (0, g) implies that
〈G,−η〉ds =329 ∫
〈G,−η〉ds = 0. Therefore330
〈G,−η〉ds =
[p,q] f
〈G,−η〉ds +
[Π(q),Π(p)]c
〈G,−η〉ds. (4.2)
In [p, q] f , the outer normal vector es parallel to −∇ f (z) = −
fx(z), fy(z)
. Then, for all z ∈ [p, q] f331
we obtain that −η(z) =
∇ f (z)
||∇ f (z)||
and G(z) = F(z) because [p, q] f ⊂ { f = f (p)}. Thus332
[p,q] f
〈G,−η〉ds =
[p,q] f
〈F,∇ f 〉dt.
Similarly, in [Π(q),Π(p)]c we have −η(z) = (0, 1) thus333
[Π(q),Π(p)]c
〈G,−η〉ds =
g(α, c)dα = −
g(α, c)dα.
Therefore, (4.2) and (4.1) prove the assertion. �334
In order to conclude the proof of this lemma we consider D(C) as in the last assertion joint335
to the construction of its precedent paragraph. Since the complement of S is a total measure set,336
gy(x, y)dx ∧ dy =
g(mα) − g(α, c)
Π(pα)=α
g(qα) − g(pα)
Thus, the formulae in the assertion implies that337
[p,q] f
〈F,∇ f 〉dt =
g(mα)dα +
Π(pα)=α
g(qα) − g(pα)
dα. (4.3)
But,338 ∫
[p,q] f
〈F,∇ f 〉dt =
[p,q] f
〈X,∇ f 〉dt − f (p)
[p,q] f
and the property (a.3) of the precedent paragraph to Assertion 1 shows that g(qα) − g(pα) ≥ 0.339
Therefore, since g(mα) ≥ g(p), (4.3) implies that340
[p,q] f
〈X,∇ f 〉dt − f (p)
[p,q] f
fxdt ≥
g(mα)dα ≥ g(p)
b − a
and concludes this proof.341
By applying the methods of the last proof give us the next:342
Lemma 6. Let X = ( f , g) : R2 \ Dσ → R2 be a map with Spc(X) ∩ [0,+∞) = ∅. Consider343
L−q ⊂ { f = f (q)} and Π(x, y) = x. If the oriented compact arc [p, q] f ⊂ L
q , and Π([p, q] f ) =344
[Π(q),Π(p)] ⊂ (σ,+∞) with Π(q) < Π(p). Then345
[p,q] f
〈X,∇ f 〉dt ≥ f (q)
[p,q] f
fxdt + g(p)
Π(p) − Π(q)
Proof. As a = Π(p) < Π(q) = b, we again consider the null set S ⊂ Π([p, q] f ) given by the346
critical values of Π restricted to [p, q] f (see [17]). Similarly, for every α ∈ Π([p, q] f ) = [a, b] we347
define ŷα = sup{y ∈ R : (y, α) ∈ [p, q] f } and mα = (α, ŷα). Therefore, Remark 5 shows that for348
each α < S the finite set
−1(α) ∩ [p, q] f
\ {m̃α} splits into pairs {pα = (α, dα), qα = (α, cα)}349
satisfying: (i) cα < dα, (ii) the oriented arc [pα, qα] f ⊂ {x ≥ α}, and (iii) g(qα) > g(pα).350
Take D(C) the boundary of which is the closed curve C ⊂ Π−1(b) ∪ [p, q] f ∪ Π
−1(a) ∪351
[Π(p),Π(q)]c, where [Π(p),Π(q)]c = {(α, c) : a ≤ α ≤ b} for some c < inf{y : (x, y) ∈ [p, q] f }.352
By using the Green’s formulae with the map z 7→ (0, g(z)) and the compact disk D(C) we have353
that354 ∫
gy(x, y)dx ∧ dy =
[p,q] f
〈F̃,∇ f 〉dt −
g(α, c)dα,
where F̃(x, y) =
f (x, y) − f (q), g(x, y)
. Since355
gy(x, y)dx ∧ dy +
g(α, c)dα =
g(mα)dα +
Π(pα)=α
g(qα) − g(pα)
and356 ∫
[p,q] f
〈F̃,∇ f 〉dt =
[p,q] f
〈X,∇ f 〉dt − f (q)
[p,q] f
fxdt,
the last property (iii) implies357
[p,q] f
〈X,∇ f 〉dt − f (q)
[p,q] f
fxdt ≥
g(mα)dα.
This concludes the proof because g(mα) ≥ g(p) shows
g(mα)dα ≥ g(p)(b − a).358
5. Hurwitz vector fields359
This section concludes with the proof of the main theorem. The essential goal of the next360
proposition is to prove the fact that our eigenvalue assumption ensures the non-existence of361
unbounded half–Reeb components. It is obtained by using the preparatory results of the previous362
section. With this fact Theorem 2 is just obtained by applying our previous papers [13, 15].363
Proposition 2 (Main). Let X = ( f , g) : R2 \ Dσ → R2 be a differentiable map, where σ > 0 and364
Dσ = {z ∈ R
2 : ||z|| ≤ σ}. Suppose that X is Hurwitz: Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Then365
(i) Any half-Reeb component of either F ( f ) or F (g) is a bounded subset of R2.366
(ii) There are s ≥ σ and a globally injective local homeomorphism X̃ = ( f̃ , g̃) : R2 → R2367
such that X̃ and X coincide on R2 \ Ds. Moreover, F ( f ) and F (g) have no half-Reeb368
components.369
Proof. In the proof of (i), we only consider one case. Suppose by contradiction that the foliation,370
given by the level curves has an unbounded half–Reeb component. By [4, Proposition 1.5], there371
exists a half–Reeb component of F ( f ) the projection of which is an interval of infinite length.372
Thus,373
PSfrag replacements
∇ f (z)
γ(0) ⊂ { f = 0}
γ(1) ⊂ { f = 0}
mα ∈ Γ
α[γ(s), γ(ϕ(s))] f
psα = (asα , dsα) ∈ {g = g(γ(0))}
qsα = (asα , csα)
Figure 3: Half-Reeb componet A, proof of Proposition 2
(a) there are â0 > σ and A, a half–Reeb component of F ( f ) such that [â0,+∞) ⊂ Π(A),374
and the vertical line Π−1(â0) intersects (transversally) both non–compact edges ofA. Here375
Π(x, y) = x.376
A half-Reeb component of a fixed foliation contain properly other half-Reeb components,377
and they are topologically equivalent. In this context, the component is stable under perturbations378
on their compact face as long as the perturbed arc is also a compact face of some component.379
Therefore, without lost of generality, we may assume that nearby its endpoints, the compact edge380
of is made up of arcs of F (g). In this way, there exist 0 < ã < 12 and an injective, continuous381
curve γ : (−ã, 1 + ã)→ A such that382
(b.1) γ([0, 1]) is a compact edge of A such that both non–compact edges L+
γ(0) and L
γ(1) are383
contained in a half–plane {x ≥ â}, for some â > σ.384
(b.2) The images γ
(−ã, ã)
and γ
(1− ã, 1+ ã)
are contained in some leaves of F (g) such that385
γ(1 − ã, 1 + ã)
< infΠ
γ(−ã, ã)
(b.3) For some 0 < δ < ã4 there exists a orientation reversing injective function ϕ : [−2δ, 2δ]→387
(1− ã, 1+ ã) with ϕ(0) = 1 such that f
γ(ϕ(s))
. Furthermore, if s ∈ (0, 2δ] then388
ϕ(s) ∈ (1− ã, 1) and there exists an oriented compact arc of trajectory [γ(s), γ(ϕ(s))] f ⊂ A389
of F ( f ), connecting γ(s) with γ(ϕ(s)).390
(b.4) For some 0 < δ < ã4 , small enough if s ∈ (0, δ] and γ(s) = (as, ds) then there exists cs such391
that qs = (as, cs) belongs to the open arc
γ(s), γ(ϕ(s))
f ⊂ [γ(s), γ(ϕ(s))] f .392
Lemma 4 implies that393
(c) for every s ∈ (0, δ] as in (b.4), cs < ds.394
The eigenvalue condition is invariant under addition of constant vectors to maps, therefore395
we can assume that396
(d) g
> 0, f = 0 over both non-compact edges ofA and f > 0 in the interior ofA.397
SinceA is the union of an increasing sequence of compact sets bounded by the compact edge398
and a compact segment of leaf. Then, from our selection of the compact edge we have that for399
every α > Π
≥ inf{as : s ∈ (0, δ]}, large enough there exists sα ∈ (0, δ] as in (b.4) such that400
the compact arc [γ(sα), γ(ϕ(sα))] f projects over (−∞, α], meeting α (Figure 3). More precisely,401
α = sup
Π(p) : p ∈ [γ(sα), γ(ϕ(sα))] f
This defines a closed curve Γ−α contained in Π
−1(asα) ∪ [γ(sα), γ(ϕ(sα))] f . If [qsα , psα]asα ⊂402
−1(asα) is the vertical segment connecting qsα , of (b.4) with psα = γ(sα), then this clock wise403
oriented curve satisfies404
α = [qsα , psα]asα ∪ [psα , qsα] f and Π(Γ
α) = [asα , α], (5.1)
where the oriented compact arc [psα , qsα] f ⊂ [γ(sα), γ(ϕ(sα))] f .405
We select and fix mα ∈ Γ
−1(α), by using Lemma 5 and Lemma 6, respectively we obtain406
[psα ,mα] f
〈X,∇ f 〉dt ≥ f (psα)
[psα ,mα] f
fxdt + g(psα)(x − asα),
and407 ∫
[mα ,qsα ] f
〈X,∇ f 〉dt ≥ f (qsα)
[mα ,qsα ] f
fxdt + g(mα)(x − asα).
Therefore, by adding we conclude408
[psα ,qsα ] f
〈X,∇ f 〉dt ≥ f (psα )
[psα ,qsα ] f
fxdt + g(psα)(x − asα) (5.2)
because g in increasing along [psα , qsα] f ⊂ { f = f (psα)} and g(mα) ≥ g(psα) = g(γ(0)) > 0.409
(e.1) We claim that, the closed curves Γ−α , given in (5.1) define the following functions410
∣∣∣∣∣∣∣
[qsα ,psα ]asα
∣∣∣∣∣∣∣
and α 7→
∣∣∣∣∣∣
[qsα ,psα ] f
∣∣∣∣∣∣ ,
they are bounded, when α varies in some interval of infinite length contained on [â0,∞).411
Furthermore,412
f (psα )
[qsα ,psα ] f
fx = 0.
In fact, by a perturbation in the compact face if it is necessary, it is not difficult to prove that413
there is some half-Reeb component Ã of F ( f ) such that Ã ⊃ A, their boundaries satisfy ∂Ã \414
{compact face} ⊃ ∂A \ {compact face} and Ã ⊃ [qsα , psα]sα , for all sα. Since the image f (Ã) ⊂415
f (compac face), the function f is bounded in the closure of Ã, which contain the compact set416
∪sα [qsα , psα]asα . Consequently,417
∣∣∣∣∣∣∣
[qsα ,psα ]asα
∣∣∣∣∣∣∣
is bounded, in some interval of infinite length contained on [â0,∞). In order to prove the second418
part, we apply the Green’s formulae to the map z 7→ (1, 0) in the compact disk D(Γ−α). Since the419
trace is cero, we obtain420
∣∣∣∣∣∣
[qsα ,psα ] f
∣∣∣∣∣∣ = arc length of [qsα , psα]asx .
By compactness we conclude. The last part is directly obtained by using (5.1) and psα = γ(sα)421
because the continuity of the foliation F ( f ) and (d) imply that422
psα = γ(0) ∈ { f = 0}.
Therefore (e.1) holds.423
(e.2) We claim that,424
if α→ +∞ then
〈X, ηiα〉dt→ +∞,
where −ηiα is a outer normal vector of the close wise oriented curve Γ
α .425
In the compact arc [psα , qsα] f ⊂ Γ
α , the vector η
α is parallel to ∇ f . Thus (5.2) and (e.1) imply426
that427 ∮
〈X, ηiα〉dt ≥ A + g(psα)(α − a),
where the constant A is independent of α and a = min{as : s ∈ (0, δ]}. Since (d) and (b.2) imply428
that g(psα) = g(γ(0)) > 0, it is not difficult to obtain (e.2).429
By (e.2) we select some α̃ > a such that Γ−α satisfies
〈X, ηiα〉dt > 0. By using the Green’s430
formulae with the map X = ( f , g), the assumptions over the eigenvalues i.e 0 > Trace(DXz),431
imply that432
〈X, ηiα〉dt > 0.
This contradiction concludes the proof of part (i).433
In order to obtain (ii), we apply Theorem 3 because X satisfies its conditions. This gives the434
existence of the pair (X̃, s) with s ≥ σ and X̃ = ( f̃ , g̃) : R2 → R2 a globally injective local435
homeomorphism such that X̃ and X coincide on R2 \ Ds. Furthermore, the last property in (ii) is436
obtained as a direct application of Corollary 2. Therefore, this proposition holds.437
Now we prove our main result438
5.1. Proof of Theorem 2439
By Proposition 2, there exists a globally injective local homeomorphism X̃ : R2 → R2440
such that X̃ and X coincide on some R2 \ Ds1 , with s1 ≥ σ. In particular, the restriction X| :441
2 \ Ds1 → R
2 is injective. In order to shown the existence of the differentiable extension,442
consider v = −X̃(0) joint to the globally injective map X̃ + v. In this context, we can apply the443
arguments of [15, Theorem 11]. Thus there are s̃ > s1 ≥ σ and a global differentiable vector444
field Y : R2 → R2 such that445
(a.1) R2 \ Ds̃ ∋ z 7→ Y(z) = (X + v)(z) is also injective and Y(0) = 0.446
(a.2) The map z 7→ Trace(DYz) is Lebesgue almost–integrable in whole R
2 ([15, Lemma 7]).447
(a.3) The index I(X + v) is a well-defined number in [−∞,+∞) ([15, Corollary 13]).448
Thus, there exist the index of X at infinity, I(X) = I(X + v). Therefore, X̂ = Y − v is the global449
differentiable extension of X| : R2 \ Ds̃ → R
2 and the pair (X̂, s̃) satisfies the definition of the450
index of X at infinity. This concludes the proof of (i).451
To prove the first part of (ii) we refer the reader to [4, Lemma 3.3]. Furthermore, since452
Trace(D(X + ṽ)) = Trace(DX) < 0, for every constant vector ṽ ∈ R2 we obtain that453
(b.1) Given a constant ṽ ∈ R2, the vector field X + ṽ generates a positive semi-flow on R2 \ Dσ.454
An immediate consequence of (i) is that: if X is Hurwitz, then outside a larger disk both X and455
Y have no rest points. In addition, by (a.1) the Hurwitz vector field Y has no periodic trajectory γ456
with D(γ) contained in R2 \Dσ. As Trace(DY) = Trace(DX) < 0 by Green’s Formulae Y admits457
at most one periodic trajectory, say γ, such that D(γ) ⊃ Dσ. Consequently458
(b.2) There exit s > s̃ such that Y satisfies (a.1), (a.2) and R2 \ Ds is free of rest points and459
periodic trajectories of Y.460
Under these conditions (b.1) and [15, Theorem 26] imply that:461
(b.3) For every r ≥ s there exist a closed curve Cr transversal to Y contained in the regular set462
2 \Dr. In particular, D(Cr) contains Dr and Cr has transversal contact to each small local463
integral curve of Y at any p ∈ Cr.464
Moreover, [15, Theorem 28] shown that:465
(b.4) The point at infinity of the Riemann Sphere R2 ∪ {∞} is either an attractor or a repellor of466
X + v : R2 \ Ds → R
2. More specifically, if I(X) < 0 (respectively I(X) ≥ 0), then∞ is a467
repellor (respectively an attractor) of the vector field X + v.468
Therefore, (ii) holds and concludes the proof of Theorem 2. �469
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	1 Introducion
	2 Statements of the results
	2.1 Description of the proof of Theorem 2
	3 Local diffeomorphisms that are injective on unbounded open sets
	3.1 Avoiding tangent points
	3.2 Minimal number of internal tangent points
	3.3 Extending maps to topological embeddings
	4 Maps free of positive eigenvalues
	5 Hurwitz vector fields
	5.1 Proof of Theorem 2
