Asymptotic stability at infinity for bidimensional Hurwitz vector fields

J. Differential Equations 255 (2013) 1050–1066

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Journal of Differential Equations www.elsevier.com/locate/jde

Asymptotic stability at infinity for bidimensional Hurwitz vector fields ✩ Roland Rabanal 1 Departamento de Ciencias, Pontificia Universidad Católica del Perú, Av. Universitaria 1801, Lima 32, Peru

a r t i c l e

i n f o

Article history: Received 16 January 2013 Revised 24 April 2013 Available online 22 May 2013 MSC: primary 37E35, 37C10 secondary 26B10, 58C25 Keywords: Injectivity Reeb component Asymptotic stability Planar vector fields

a b s t r a c t Let X : U → R2 be a differentiable vector field. Set Spc( X ) = {eigenvalues of D X (z): z ∈ U }. This X is called Hurwitz if Spc( X ) ⊂ { z ∈ C: (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 a 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 X˜ : R2 → R2 , a topological embedding such that X˜ equals X in the complement of some compact subset of U . © 2013 Elsevier Inc. All rights reserved.

1. Introduction A basic example of non-discrete dynamics on the Euclidean space is given by a linear vector field. This linear system is infinitesimally hyperbolic if every eigenvalue has nonzero real part, and it has well-known properties [16,3]. For instance, when the real part of its eigenvalues are negative (Hurwitz matrix), the origin is a global attractor rest point. In the nonlinear case, there has been a great interest in the local study of vector fields around their rest points [6,27,7,28]. However, in order to describe a global phase-portrait, as in [22,25,5,8,9] it is absolutely necessary to study its behavior in a neighborhood of infinity [19].

✩

1

This paper was written when the author served as an Associate Fellow at ictp-Italy. E-mail address: [email protected]. The author was partially supported by pucp-Peru (dgi: 70242-2010).

0022-0396/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2013.04.037

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The Asymptotic Stability at Infinity has been investigated with a strong influence of [18], where Olech showed a connection between stability and injectivity (see also [10,4,11,26,22]). This problem was also researched in [19,12,14,15,23,1]. In [12], Gutierrez and Teixeira consider C 1 -vector fields Y : R2 → R2 , the linearizations of which satisfy (i) det( D Y ( z)) > 0 and (ii) Trace( D Y ( z)) < 0 in a neighborhood of infinity. By using [9], they prove that if Y has a rest point and the Index  I (Y ) = Trace ( D Y ) < 0 (resp. I (Y )  0), then Y is topologically equivalent to z → −z that is “the infinity is a repellor” (resp. to z → z that is “the infinity is an attractor”). This Gutierrez–Teixeira’s paper was used to obtain the next theorem, where Spc(Y ) = {eigenvalues of D Y ( z): z ∈ R2 \ D σ }, and (z) is the real part of z ∈ C. Theorem 1 (Gutierrez–Sarmiento). Let Y : R2 \ D σ → R2 be a C 1 − map, where R2 :  z  σ }. The following is satisfied:

σ > 0 and D σ = {z ∈

(i) If for some ε > 0, Spc(Y ) ∩ (−ε , +∞) = ∅. Then there exists s  σ such that the restriction Y | : R2 \ D s → R2 is injective. (ii) If for some ε > 0, the spectrum Spc(Y ) is disjoint of the union (−ε , 0] ∪ { z ∈ C: ( z)  0}. Then there exist p 0 ∈ R2 such that the point ∞ of the Riemann Sphere R2 ∪ {∞} is either an attractor or a repellor of z = Y ( z) + p 0 . Theorem 1, given in [14], has been extended to differentiable maps X : R2 \ D σ → R2 in [13,15]. In both papers the eigenvalues also avoid a real open neighborhood of zero. In [23], the author examine the intrinsic relation between the asymptotic behavior of Spc( X ) and the global injectivity of the local diffeomorphism given by X . He uses Y θ = R θ ◦ Y ◦ R −θ , where R θ is the linear rotation of angle θ ∈ R, and (motivated by [11]) introduces the so-called B-condition [24,26], which claims: for each θ ∈ R, there does not exist a sequence (xk , yk ) ∈ R2 with xk → +∞ such that Y θ ((xk , yk )) → p ∈ R2 and D Y θ (xk , yk ) has a real eigenvalue λk satisfying xk λk → 0. By using this, [23] improves the differentiable version of Theorem 1. In the present paper we prove that the condition



Spc( X ) ⊂ z ∈ C: ( z) < 0



is enough in order to obtain Theorem 1 for differentiable vector fields X : R2 \ D σ → R2 . Throughout this paper, R2 is embedded in the Riemann Sphere R2 ∪ {∞}. Thus (R2 \ D σ ) ∪ {∞} is the subspace of R2 ∪ {∞} with the induced topology, and ‘infinity’ refers to the point ∞ of R2 ∪ {∞}. Moreover given C ⊂ R2 , a closed (compact, no boundary) curve (1-manifold), D (C ) (respectively D (C )) is the compact disc (respectively open disc) bounded by C . Thus, the boundaries ∂ D (C ) and ∂ D (C ) are equal to C , homeomorphic to ∂ D 1 = { z ∈ R2 :  z = 1}. 2. Statement of the results For every σ > 0 let D σ = { z ∈ R2 :  z  σ }. Outside this compact disk we consider a differentiable vector field X : R2 \ D σ → R2 . As usual, a trajectory of X starting at q ∈ R2 \ D σ is defined as the integral curve determined by a maximal solution of the Initial Value Problem z˙ = X ( z), z(0) = q. This is a curve I q  t → γq (t ) = (x(t ), y (t )), satisfying:

• • • •

γq (0) = q; dy γq (t ) ∈ R2 \ D σ and there exist the real derivatives dx ( t ) , ( t ) ; dt dt γ˙q (t ) = ( dx (t ), dy (t )), the velocity vector field of γq at γq (t ) equals X (γq (t )), and dt dt

t varies on some open real interval containing the zero, the image of which

I q ⊂ R is the maximal interval of definition.

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We identify the trajectory γq with its image γq ( I q ), and we denote by γq+ (resp. γq− ) the positive (resp. negative) semi-trajectory of X , contained in γq and starting at q. In this way γq = γq− ∪ γq+ . Thus each trajectory has its two limit sets, α (γq− ) and ω(γq+ ) respectively. These limit sets are well defined in the sense that they only depend on the respective solution. A C 0 -vector field X : R2 \ D σ → R2 \ {0} (without rest points) can be extended to a map

Xˆ :

  2    R \ D σ ∪ ∞ , ∞ → R2 , 0

(which takes ∞ to 0) [1]. In this manner, all questions concerning the local theory of isolated rest points of planar vector fields can be formulated and examined in the case of the vector field Xˆ . For instance, if γ p+ (resp. γ p− ) is a unbounded semi-trajectory of X : R2 \ D σ → R2 starting at p ∈ R2 \ D σ with empty ω -limit (resp. α -limit) set, we say γ p+ goes to infinity (resp. γ p− comes from infinity), and it is denoted by ω(γ p+ ) = ∞ (resp. α (γ p− ) = ∞). Therefore, we may also talk about the phase portrait of X in a neighborhood of ∞, as shown [15, Proposition 29]. As in our paper [15], we say that the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is an attractor (respectively a repellor) for the continuous vector field X : R2 \ D σ → R2 if:

• There is a sequence of closed curves, transversal to X and tending to infinity. That is for every r  σ there exists a closed curve C r such that D (C r ) contains D r and C r has transversal contact to each small local integral curve of X at any p ∈ C r . • For some C s with s  σ , all trajectories γ p starting at a point p ∈ R2 \ D (C s ) satisfy ω(γ p+ ) = ∞ that is γ p+ goes to infinity (respectively α (γ p− ) = ∞ that is γ p− comes from infinity). We also recall that I ( X ), the index of X at infinity is the number of the extended line [−∞, +∞] given by

 I( X) =

Trace( D  X ) dx ∧ dy ,

R2

where  X : R2 → R2 is a global differentiable vector field such that:

• In the complement set of some compact disk D s with s  σ , both X and  X coincide. • The map z → Trace( D  X z ) is Lebesgue almost-integrable on whole R2 , in the sense of [15]. This index is a well-defined number in [−∞, +∞], and it does not depend on the pair ( X , s) as shown [15, Lemma 12]. Definition 1. The differentiable vector field (or map) X : R2 \ D σ → R2 is called Hurwitz if every eigenvalue of the Jacobian matrices has negative real part. This means that its spectrum Spc( X ) = {eigenvalues of D X (z): z ∈ R2 \ D σ } satisfies Spc( X ) ⊂ { z ∈ C: (z) < 0}. We are now ready to state our result. Theorem 2. Let X : R2 \ D σ → R2 be a differentiable vector field (or map), where R2 :  z  σ }. Suppose that X is Hurwitz: Spc( X ) ⊂ { z ∈ C: (z) < 0}. Then:

σ > 0 and D σ = {z ∈

(i) There are s  σ and a globally injective local homeomorphism X˜ : R2 → R2 such that X˜ and X coincide on R2 \ D s . Moreover, the restriction X | : R2 \ D s → R2 is injective, and it admits a global differentiable X such that the pair ( X , s) satisfies the definition of the index of X at infinity, and this index extension  I ( X ) ∈ [−∞, +∞).

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(ii) For all p ∈ R2 \ D σ , there is a unique positive semi-trajectory of X starting at p. Moreover, for some v ∈ R2 , the point at infinity ∞ of the Riemann Sphere R2 ∪ {∞} is an attractor (respectively a repellor) for the vector field X + v : R2 \ D s → R2 as long as the well-defined index I ( X )  0 (respectively I ( X ) < 0). The map X˜ of Theorem 2 is not necessarily a homeomorphism. This X˜ is a topological embedding, the image of which may be properly contained in R2 . Furthermore, if A : R2 → R2 is an arbitrary invertible linear map, then Theorem 2 applies to the map A ◦ X ◦ A −1 . Theorem 2 improves the main results of [13,15]. Item (i) complements the injectivity work of [13] (see also [23,14]), where the authors consider the assumption Spc( X ) ∩ (−ε , +∞) = ∅ (as in Theorem 1). Item (ii) generalizes [15], where the authors utilize the second condition of Theorem 1. In our new assumption, the negative eigenvalues can tend to zero. 2.1. Brief description of the proof of Theorem 2 Since the Local Inverse Function Theorem is true, a map X = ( f , g ) : R2 \ D σ → R2 as in Theorem 2 is a local diffeomorphism. Thus, the level curves { f = constant} make up a C 0 -foliation F ( f ) the leaves of which are differentiable curves. The restriction of the other submersion g to each of these leaves is strictly monotone, hence both foliations F ( f ) and F ( g ) are (topologically) transversal to each other. We orient F ( f ) in agreement that if L p ( f ) is an oriented leaf of F ( f ) thought the point p, then the restriction g | : L p ( f ) → R is an increasing function in conformity + with the orientation of L p ( f ). We denote by L + p ( f ) = { z ∈ L p ( f ): g ( z)  g ( p )} (resp. L p ( g ) = { z ∈ − − L p ( g ): f ( z)  f ( p )}) and L p ( f ) = { z ∈ L p ( f ): g ( z)  g ( p )} (resp. L p ( g ) = { z ∈ L p ( g ): f ( z)  f ( p )}) the respective positive and negative half-leaf of F ( f ) (resp. F ( g )). Thus L q ( f ) = L q− ( f ) ∪ L q+ ( f ) and L q− ( f ) ∩ L q+ ( f ) = {q}. In this context, the nonsingular vector fields

X f = (− f y , f x )

and

X˜ g = ( g y , − g x ),

(2.1)

given by the partial derivatives are tangent to L p ( f ) and L p ( g ), respectively. We need the following definition [14,4]. Let h0 (x, y ) = xy and consider the set





B = (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 say that A (in the domain of X ) is a half-Reeb component for F (h) if there is a homeomorphism H : B → A which is a topological equivalence between F (h)|A and F (h0 )| B such that (Fig. 4):

• 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 the point H (1, 1). This section is called the compact edge of A. • Both segments {(x, y ) ∈ B: x = 0} and {(x, y ) ∈ B: y = 0} are sent by H onto full half-leaves of F (h). These half-leaves of F (h) are called the non-compact edges of A. Observe that A may not be a closed subset of R2 , and H does not need to be extended to infinity. Section 3 gives new results on the foliations induced by a local diffeomorphism X = ( f , g ) : R2 \ D σ → R2 (see also Proposition 1). Theorem 3 implies that the conditions

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 \ D s can be extended to an injective map X˜ : R2 → R2 . Section 4 presents some preliminary results on maps with Spc( X ) ∩ [0, +∞) = ∅. Section 5 concludes with the proof of Theorem 2. The main step is given in Proposition 2, which implies that Hurwitz maps satisfy (2.2). Therefore, Theorem 2 is obtained by using this Proposition 2 and some previous work [13,15].

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3. Local diffeomorphisms that are injective on unbounded open sets Let X = ( f , g ) : R2 \ D σ → R2 be an orientation preserving local diffeomorphism, that is det( D X ) > 0. Next subsection gives preparatory results about F ( f ) in order to obtain that X is injective on topological half planes (see Proposition 1). Subsection 3.3 presents a condition under which X is injective outside some compact set. 3.1. Avoiding tangent points Let C ⊂ R2 \ D σ be a closed curve surrounding the origin. We say that the vector field X : R2 \ D σ → R2 has contact (resp. tangency with; resp. transversal to; etc.) with C at p ∈ C if for each small local integral curve of X at p has such property. Definition 3. A closed curve C ⊂ R2 \ D σ is in general-position with F ( f ) if there exists a set T ⊂ C , at most finite such that:

• F ( f ) is transversal to C \ T . • F ( f ) has a tangency with C at every point of T . • A leaf of F ( f ) can meet tangentially C at most at one point. We denote by GP ( f , s) the set of all closed curves C ⊂ R2 \ D σ in general-position with F ( f ) such that D s ⊂ D (C ). If C ⊂ R2 \ D σ is in general-position, we denote by ne (C , f ) (resp. ni (C , f )) the number of external (resp. internal) tangent points of F ( f ) with C . Here, external (resp. internal) means the existence of a small open interval ( p˜ , q˜ ) f ⊂ L p ( f ) such that the intersection set ( p˜ , q˜ ) f ∩ C = {tangent point} and ( p˜ , q˜ ) f ⊂ R2 \ D (C ) (resp. ( p˜ , q˜ ) f ⊂ D (C )). Remark 1. If C ⊂ R2 \ D σ is in general-position with F ( f ), there exist a, b ∈ C two different points such that f (C ) = [ f (a), f (b)]. Moreover, a and b are external tangencies because the map ( f , g ) is an orientation preserving local diffeomorphism. Since C and f (C ) are connected and C is not contained in any leaf of F ( f ), we conclude that both external tangencies are different. Corollary 1 gives important properties of the leafs passing trough a point as in Remark 1, if we select C ∈ GP ( f , σ ) with the minimal number of internal tangencies. To prove it, the next lemma is needed. Lemma 1. Let C ∈ GP ( f , σ ). Suppose that a leaf L q ( f ) of F ( f ) meets C transversally somewhere and with an external tangency at a point p ∈ C . Then L q ( f ) contains a closed subinterval [ p , r ] f which meets C exactly at { p , r } (doing it transversally at r) and the following is satisfied: (a) If [ p , r ] is the closed subinterval of C such that Γ = [ p , r ] ∪ [ p , r ] f bounds a compact disc D (Γ ) contained in R2 \ D (C ), then points of L q ( f ) \ [ p , r ] f nearby p do not belong to D (Γ ). (b) Let ( p˜ , r˜ ) and [ p˜ , r˜ ] be subintervals of C satisfying [ p , r ] ⊂ ( p˜ , r˜ ) ⊂ [ p˜ , r˜ ]. If p˜ and r˜ are close enough to p and r, respectively; then we may deform C into C 1 ∈ GP ( f , σ ) in such a way that the deformation fixes C \ ( p˜ , r˜ ) and takes [ p˜ , r˜ ] ⊂ C to a closed subinterval [ p˜ , r˜ ]1 ⊂ C 1 which is close to [ p , r ] f . Furthermore, the number of generic tangencies of F ( f ) with C 1 is smaller than that of F ( f ) with C . Proof. We refer the reader to [14, Lemma 2].

2

Corollary 1. Let C ∈ GP ( f , s). Suppose that C minimizes ni (C , f ) and f (C ) = [ f (a), f (b)], then C ∩ L a ( f ) = {a} and C ∩ L b ( f ) = {b}. Proof. We only consider the case of the point a. Assume by contradiction that the number of elements in C ∩ L a ( f ) is greater than one i.e (C ∩ L a ( f ))  2. The last condition of Definition 3 implies

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that the intersection of L a ( f ) \ {a} is transversal to C . Then there is a disk as in statement (a) of Lemma 1. By using the second part of Lemma 1 we can avoid the external tangency a and some internal tangency. This is a contradiction because ni (C , f ) is minimal. 2 Remark 2. Corollary 1 remains true if we take any external tangency (not necessarily a and b) in a closed curve C s ∈ GP ( f , s) with minimal ni (C s , f ). 3.2. Minimal number of internal tangent points An oriented leaf of F ( f ) whose distance to D σ is different from zero has unbounded half-leaves. Given any L p ( f ) with unbounded half-leaves, we denote by H + ( L p ) and H − ( L p ) the two components + + of R2 \ L p ( f ) in order that L + p ( g ) \ { p } be contained in H ( L p ). Therefore, X ( H ( L p )) is an open connected subset of the semi-plane {x > f ( p )} := {(x, y ) ∈ R2 : x > f ( p )}. Remark 3. If the image X ( H + ( L p )) is a vertical convex set, all the level curves { f = c } ⊂ H + ( L p ) are connected. Thus, the restriction X | : H + ( L p ) → X ( H + ( L p )) is a homeomorphism, and it sends every leaf of F ( f )| H + ( L p ) over vertical lines. Therefore, it is a topological equivalence between two foliations. Lemma 2. If H + ( L p ) is disjoint from D σ and the image X ( H + ( L p )) is not a vertical convex set, then H + ( L p ) contains a half-Reeb component of F ( f ). Proof. By Remark 3, some level set { f = c˜ } ⊂ H + ( L p ) is disconnected. Therefore, the result is obtained directly from [11, Proposition 1]. 2 Lemma 2 and Remark 3 hold when we consider H − ( L p ). Lemma 3. Recall that GP ( f , s) is the set of all closed curves C ⊂ R2 \ D σ in general-position with F ( f ) such that D s ⊂ D (C ). Let ηi : [σ , ∞) → N ∪ {0} be the function given by ηi (s) = ni (C s , f ) where C s ∈ GP ( f , s) minimizes the number of internal tangent points with F ( f ). The following statements hold: (a) The function ηi is nondecreasing. (b) If ηi is bounded then, there exist s0 ∈ [σ , ∞) such that ηi (s)  ηi (s0 ) for all s ∈ [σ , ∞). (c) Set f (C s0 ) = [ f (a), f (b)]. Suppose that F ( f ) has a half-Reeb component A whose image f (A) is disjoint from ( f (a), f (b)), then such s0 ∈ [σ , ∞) is not a maximum value of the function ηi . Proof. As C s+1 also belongs to GP ( f , s) we have that ηi (s)  ni (C s+1 , f ). Therefore (a) is true. To prove the second part, we introduce the set A s = {n ∈ N ∪ {0}: n  ni (C s , f ) = ηi (s)}, for every s  σ . Consequently: ηi is bounded if and only if sσ A s = ∅. Therefore, the first element of



A s = ∅ is the bound of

ηi (s0 ) in the second statement, and proves (b). We shall have established (c) if we prove that there is some C s0 +ε with ε > 0 such that ηi (s0 )+ 1 

sσ

ni (C s0 +ε , f ). To this end, we select s > s0 large enough for which C s is enclosing D (C s0 ) ∪ Γ where Γ is the compact edge of A. Since, this C s intersects both leaves L p ( f ) and L q ( f ) where p and q are the endpoints of Γ , we obtain that ni (C s , f ) is greater than ni (C s0 , f ) = ηi (s0 ). Therefore, for some ε > 0 there is C s0 +ε such that ηi (s0 ) + 1  ni (C s0 +ε , f ). This proves (c). 2

Proposition 1. Let X = ( f , g ) : R2 \ D σ → R2 be a map with det( D X ) > 0. Consider C s and ηi as in Lemma 3. If C s0 satisfies that ni (C s , f )  ni (C s0 , f ) for all s ∈ [σ , ∞) and f (C s0 ) = [ f (a), f (b)]. Then, for each p ∈ {a, b} at least one of the restrictions of X to H + ( L p ) or H − ( L p ) is a globally injective map, in agreement that the domain of this restriction is in the complement of D (C s0 ).

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Proof. We only consider the case p = a. Suppose that H + ( L a ) is contained in the complement of D (C s0 ). From Remark 3 it is sufficient to prove that X ( H + ( L a )) is vertical convex. Suppose by contradiction that it is false. Then Lemma 2 implies that there is a half-Reeb component A ⊂ H + ( L a ). By using statement (c) of Lemma 3 this s0 is not a maximum value of the function η i . This contradiction with our selection of the circle C s0 conclude the proof. 2 3.3. Extending maps to topological embeddings The next theorem implies the injectivity at infinity of a map, and it is obtained by using the methods, ideas and arguments of [13]. We only give the proof, in the case of continuously differentiable maps. Theorem 3. Let X = ( f , g ) : R2 \ D σ → R2 be a differentiable local homeomorphism with det( D X ) > 0. Suppose that Spc( X ) ∩ [0, +∞) = ∅, and each half-Reeb component of either F ( f ) or F ( g ) is bounded. Then there exist s  σ such that the restriction X | : R2 \ D s → R2 can be extended to a globally injective local homeomorphism X˜ = ( ˜f , g˜ ) : R2 → R2 . Proof. We can apply the results of [16, pp. 166–174] to the continuous vector field X f and obtain that for each closed curve C ∈ GP ( f , σ ) the Index of X f along C , denoted by Ind( X f ; C ), satisfies

Ind( X f ; C ) =

2 − ne ( f , C ) + n i ( f , C ) 2

.

If X f is discontinuous, we proceed as in [13] by using the index of the foliation F ( f ) which also satisfies this formulae. (a.1) We claim that ne ( f , C ) = ni ( f , C ) + 2, for all C ∈ GP ( f , σ ). Suppose that (a.1) is false: there is C˜ 1 ∈ GP ( f , σ ) with Ind( X f ; C˜ 1 ) = 0. Thus, for some point in

C the vector X f ( p ) = (− f y ( p ), f x ( p )) is vertical. More precisely, we can obtain p ∈ C˜ 1 such that f y ( p ) = 0 and f x ( p ) > 0. This is a contradiction with the eigenvalue assumptions because f x ( p ) ∈ Spc( X ) ∩ (0, +∞). (If X f is discontinuous, we refer the reader to [13, Proposition 3.1], where proves that the index of the foliation F ( f ) is zero). Therefore, (a.1) holds.

˜1

(a.2) We claim that if C σ ⊂ R2 \ D s minimizes ni ( f , C σ ), then every internal tangency in C σ gives a half-Reeb component. For every internal tangency q ∈ C σ we consider the forward Poincaré map T : [ p , q)σ ⊂ C σ → C σ induced by the oriented F ( f ) (if T : (q, r ]σ ⊂ C σ → C σ the proof is similar) where [ p , q)σ ⊂ C σ is the maximal connected domain of definition of T on which this first return map is continuous. If the open arc L + p ( f ) \ { p } intersects C σ we apply Lemma 1, so we can deform C σ in a new circle

C˜ 1 ⊂ R2 \ D s such that the number of internal tangencies of C˜ 1 with F ( f ) is (strictly) smaller than that of C σ . This is a contradiction. Therefore L + p ( f ) \ { p } is disjoint from C σ . By using this and our selection of [ p , q)σ ⊂ C σ is not difficult to check that there is a half-Reeb component of F ( f ) whose compact edge is contained in C s , thus, we obtain (a.2). Notice that, for every circle as in (a.2) any internal tangency of this C σ gives a unbounded halfReeb component, thus by our assumptions and (a.1) we have that ni ( f , C σ ) = 0. Therefore,

(a.3) if C σ ∈ GP ( f , σ ) is as in (a.2) then ni ( f , C σ ) = 0 and ne ( f , C σ ) = 2. Moreover, F ( f ), restricted to R2 \ D (C σ ), is topologically equivalent to the foliation made up by all the vertical straight lines, on R2 \ D 1 .

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Since X has no unbounded half-Reeb component, we can use the last section of [13] (see Proposition 5.1) and obtain that the closed curve C σ of (a.3) can be deformed so that, the resulting new closed curve C has an exterior collar neighborhood U ⊂ R2 \ D (C ) such that: (b) X (C ) is a non-trivial closed curve, X (U ) is an exterior collar neighborhood of X (C ) and the restriction X | : U → X (U ) is a homeomorphism. By Schoenflies Theorem [2] the map X | : C → X (C ) can be extended to a homeomorphism X 1 : D (C ) → D ( X (C )). We extend X : R2 \ D (C ) → R2 to X˜ = ( ˜f , g˜ ) : R2 → R2 by defining X˜ | D (C ) = X 1 .

Thus X˜ | : U → X (U ) is a homeomorphism and U (resp. X (U )) is an exterior collar neighborhood of C (resp. X (C )). Consequently, X˜ is a local homeomorphism and F ( ˜f ) is topologically equivalent to the foliation made up by all the vertical straight lines. The injectivity of X˜ follows from the fact that F ( ˜f ) in trivial [4, Proposition 1.4]. This concludes the proof. 2

Corollary 2. Suppose that X satisfies Theorem 3. Then the respective extension X˜ = ( ˜f , g˜ ) : R2 → R2 of X is a globally injective local homeomorphism the foliations of which, F ( ˜f ) and F ( g˜ ) have no half-Reeb components. Proof. We reefer the reader to declaration (a.3) in the proof of Theorem 3.

2

Corollary 3. Suppose that X = ( f , g ) : R2 \ D σ → R2 is an orientation preserving local diffeomorphism. Then the foliation F ( f ) (resp. F ( g )) has at most countably many half-Reeb components. Proof. A half-Reeb component has a tangency with some C n of Lemma 3, where n ∈ N is large enough. We conclude, since the closed curves has at most a finite number of such tangencies. 2 Remark 4. By using a smooth embedding ( f , g ) : R2 → R2 , the authors of [10, Proposition 1] prove the existence of foliations F ( f ) which have infinitely many half-Reeb components. 4. Maps free of positive eigenvalues We present some properties of a map the spectrum of which is disjoint of [0, +∞). These results will be used in Section 5 to proving the first part of Theorem 2. In this context, we consider F ( f ) and their trajectories L q = L q ( f ), L q+ = L q+ ( f ) and L q− = L q− ( f ). Lemma 4. Let X = ( f , g ) : R2 \ D σ → R2 be a differentiable local homeomorphism. Suppose that the spec2 trum Spc( X ) ∩ [0, +∞) = ∅ and p = (a, c ). Then the intersection of L + p with the vertical ray {(a, y ) ∈ R : y  c } is the one point set { p }. 2 Proof. Assume, by contradiction, that L + p \ { p } intersects {(a, y ) ∈ R : y  c }. We take d > c the + smallest value such that q = (a, d) ∈ L p (see Fig. 1a). We only consider the case in which the compact arc [ p , q] f ⊂ L + p satisfies Π([ p , q ] f ) = [a, a0 ] with a < a0 and Π(x, y ) = x (in the other case, Π([ p , q] f ) = [a0 , a], a0 < a the argument is similar). Therefore, if we take the vertical segment [ p , q]a = {(a, y ): c  y  d} joint to the open disk D (C ) bounded by the closed curve C = [ p , q] f ∪ [ p , q]a . We meet two possible cases: The first one is that D (C ) ∩ D σ = ∅. We select the point (a0 , c 0 ) ∈ [ p , q] f in order that c 0 will be the smallest value of the compact set Π −1 (a0 ) ∩ [ p , q] f . Let R be the closed region bounded by the union of {(a, y ): y  c }, {(a0 , y ): y  c 0 } and [ p , q0 ] f ⊂ [ p , q] f where q0 = (a0 , c 0 ). As c 0 = inf{ y ∈ Π −1 (a0 ): (a0 , y ) ∈ [ p , q] f } the compact arc [ p , q] f is tangent to the vertical line Π −1 (a0 ) at the point (a0 , c 0 ). Thus, X f (a0 , c 0 ) is vertical, and so f y (a0 , c 0 ) = 0. This implies that f x (a0 , c 0 ) ∈ Spc( X ). By the assumptions about Spc( X ), f x (a0 , c 0 ) < 0 which in turn implies that the arc [q0 , q] f ⊂ [ p , q] f

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Fig. 1. Proof of Lemma 4.

must enter into R and cannot cross the boundary of R (see Fig. 1b). This contradicts the fact that / R. q = (a, d) ∈ The second case happens when D (C ) ∩ D σ = ∅. As [ p , q] f is not contained in D σ , either σ < a0 or σ = a0 . If σ < a0 , the vertical line x = σ meet [ p , q] f in two different points which define a closed curve as in Fig. 1a such that it bounds an open disk disjoint of D σ . We conclude by using the proof of the first case. If σ = a0 , we observe the continuous foliation F ( f ) in a neighborhood of [ p , q] f

˜ but Π([ p˜ , q˜ ] f ) ⊂ [σ , +∞). It satisfies the and meet two points p˜ = (σ , c˜ ) and q˜ = (σ , d˜ ) with c˜ < d, conditions of the first case. Therefore the lemma is proved. 2 Remark 5. Lemma 4 remains true, if we consider the negative leaf L q− starting at q = (a, d) joint to the vertical ray {(a, y ) ∈ R2 : y  d}. Lemma 5. Let X = ( f , g ) : R2 \ D σ → R2 be a map with Spc( X ) ∩ [0, +∞) = ∅. Consider L + p ⊂ { f = f ( p )} and the projection Π(x, y ) = x. If the oriented compact arc [ p , q] f ⊂ L + , and the image Π([ p , q] f ) is the p interval [Π( p ), Π(q)] ⊂ (σ , +∞) with Π( p ) < Π(q). Then



  X , ∇ f  dt  f ( p )

[ p ,q] f





f x dt + g ( p ) Π(q) − Π( p ) ,

[ p ,q] f

where  ,  denotes the usual inner product on the plane, and f x is the first partial derivative. Proof. For each α ∈ Π([ p , q] f ) = [a, b], the vertical line Π −1 (α ) intersects [ p , q] f in a non-empty compact set. So there exist yˆ α = sup{ y ∈ R: ( y , α ) ∈ [ p , q] f } and mα = (α , yˆ α ). We also define S ⊂ (a, b) as the set of critical values of the projection Π(x, y ) = x restricted to the differentiable arc [ p , q] f . By Sard’s Theorem, presented in [17, Theorem 3.3] this set S is closed and has zero Lebesgue measure. For α ∈ (a, b) \ S the set Π −1 (α ) intersects [ p , q] f in at most finitely many points and the complement set (Π −1 (α ) ∩ [ p , q] f ) \ {mα } is either empty or its cardinality is even. By Lemma 4, the order of these points in the line Π −1 (α ) oriented oppositely to the y-axis coincides with that on the oriented arc [ p , q] f (a behavior as in Fig. 1a does not exist). Therefore, for every α ∈ (a, b) \ S the set (Π −1 (α ) ∩ [ p , q] f ) \ {mα } splits into pairs { p α = (α , dα ), qα = (α , c α )} with the following three properties: (a.1) c α < dα , (a.2) the compact arc [ p α , qα ] f lies in the semi-plane {x  α } and it is oriented from p α to qα (a.3) g (qα ) > g ( p α ). Notice that the tangent vector of [ p , q] f at p α has a negative x-component: f y ( p α ) > 0, and the respective tangent vector at qα satisfies f y (qα ) < 0. Similarly, f y (mα )  0 (see (2.1)).

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Fig. 2. Here {∇ f ( z), X f ( z)} a positive basis.

Assertion Take c < inf{ y: (α , y ) ∈ [ p , q] f } and [Π( p ), Π(q)]c = {(α , c ): a  α  b}, a horizontal segment. Consider the compact set D (C ) ⊂ {(x, y ): a  x  b} the boundary of which contain [ p , q] f ∪ [Π( p ), Π(q)]c (see Fig. 2b). Suppose that C , the boundary of D (C ) is negatively oriented (clock wise). Then



b

 g y (x, y ) dx ∧ dy =

 F , ∇ f  dt −

[ p ,q] f

D (C )

g (α , c ) dα , a

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]) with the differentiable map G : z → (0, g ( z)) and the outer normal vector of C denoted by z → η( z) (unitary). By using that Trace( D G z ) = g y ( z) it follows that



 g y (x, y ) dx ∧ dy =

G , −η ds,

(4.1)

C

D (C )

where ds denotes the arc length element. If [Π(q), Π( p )]c denote [Π( p ), Π(q)]c oriented from Π(q) to Π( p ), then C = [ p , q] f ∪ B ∪ [Π(q), Π( p )]c ∪ A with A and B two oriented vertical segments. Consequently,



 G , −η ds =

C

 G , −η ds +

[ p ,q] f

B

In  A ∪ B the vector −η is horizontal i.e G , −η ds = 0. Therefore B



 G , −η ds +

[Π (q),Π ( p )]c

G , −η ds. A

η(z) = (η1 (z), 0) then G = (0, g ) implies that

 G , −η ds =

C

 G , −η ds +

[ p ,q] f

 A

G , −η ds =

 G , −η ds + [Π (q),Π ( p )]c

G , −η ds.

(4.2)

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In [ p , q] f , the outer normal vector es parallel to −∇ f ( z) = −( f x ( z), f y ( z)). Then, for all z ∈ [ p , q] f we ∇ (z)

obtain that −η( z) = ∇ f (z) and G ( z) = F ( z) because [ p , q] f ⊂ { f = f ( p )}. Thus f



 G , −η ds =

[ p ,q] f

 F , ∇ f  dt .

[ p ,q] f

Similarly, in [Π(q), Π( p )]c we have −η( z) = (0, 1) thus



a G , −η ds =

[Π (q),Π ( p )]c

b g (α , c ) dα = −

g (α , c ) dα . a

b

2

Therefore, (4.2) and (4.1) prove the assertion.

In order to conclude the proof of this lemma we consider D (C ) as in the last assertion joint to the construction of its precedent paragraph. Since the complement of S is a total measure set,



b



g y (x, y ) dx ∧ dy =



g (mα ) − g (α , c ) dα

a

D (C )





+



g (qα ) − g ( p α ) dα .

α ∈/ S Π ( p α )=α

Thus, the formulae in the assertion implies that



b  F , ∇ f  dt =

[ p ,q] f





g (mα ) dα +



g (qα ) − g ( p α ) dα .

(4.3)

α ∈/ S Π ( p α )=α

a

But,



  F , ∇ f  dt =

[ p ,q] f

  X , ∇ f  dt − f ( p )

[ p ,q] f

f x dt

[ p ,q] f

and the property (a.3) of the precedent paragraph to Assertion 1 shows that g (qα ) − g ( p α )  0. Therefore, since g (mα )  g ( p ), (4.3) implies that



b

  X , ∇ f  dt − f ( p )

[ p ,q] f

and concludes this proof.

[ p ,q] f

2

f x dt 

g (mα ) dα  g ( p )(b − a), a

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Fig. 3. Proof of Lemma 6.

By applying the methods of the last proof give us the next: Lemma 6. Let X = ( f , g ) : R2 \ D σ → R2 be a map with Spc( X ) ∩ [0, +∞) = ∅. Consider 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 ) = [Π(q), Π( p )] ⊂ (σ , +∞) with Π(q) < Π( p ). Then







 X , ∇ f  dt  f (q) [ p ,q] f



f x dt + g ( p ) Π( p ) − Π(q) .

[ p ,q] f

Proof. (See Fig. 3.) As a = Π(q) < Π( p ) = b, we again consider the null set S ⊂ Π([ p , q] f ) given by the critical values of Π restricted to [ p , q] f (see [17]). Similarly, for every α ∈ Π([ p , q] f ) = [a, b] we define yˆ α = sup{ y ∈ R: ( y , α ) ∈ [ p , q] f } and mα = (α , yˆ α ). Therefore, Remark 5 shows that for each α ∈/ S the finite set (Π −1 (α ) ∩ [ p , q] f ) \ {m˜ α } splits into pairs { p α = (α , dα ), qα = (α , cα )} satisfying: (i) c α < dα , (ii) the oriented arc [ p α , qα ] f ⊂ {x  α }, and (iii) g (qα ) > g ( p α ). Take D (C ) the boundary of which is the closed curve C ⊂ Π −1 (b) ∪ [ p , q] f ∪ Π −1 (a) ∪ [Π(q), Π( p )]c , where [Π(q), Π( p )]c = {(α , c ): a  α  b} for some c < inf{ y: (x, y ) ∈ [ p , q] f }. By using the Green’s formulae with the map z → (0, g ( z)) and the compact disk D (C ) we have that





 F˜ , ∇ f  dt −

g y (x, y ) dx ∧ dy = [ p ,q] f

D (C )

b g (α , c ) dα , a

where F˜ (x, y ) = ( f (x, y ) − f (q), g (x, y )). Since



b g y (x, y ) dx ∧ dy +

b g (α , c ) dα =

a

D (C )





g (mα ) dα + a

α ∈/ S Π ( p α )=α

and

 [ p ,q] f

the last property (iii) implies

 F˜ , ∇ f  dt =



[ p ,q] f

  X , ∇ f  dt − f (q) [ p ,q] f



g (qα ) − g ( p α ) dα

f x dt ,

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b

  X , ∇ f  dt − f (q)

[ p ,q] f

f x dt 

[ p ,q] f

This concludes the proof because g (mα )  g ( p ) shows

g (mα ) dα . a

b a

g (mα ) dα  g ( p )(b − a).

2

5. Hurwitz vector fields This section concludes with the proof of the main theorem. The essential goal of the next proposition is to prove the fact that our eigenvalue assumption ensures the non-existence of unbounded half-Reeb components. It is obtained by using the preparatory results of the previous section. With this fact Theorem 2 is just obtained by applying our previous papers [13,15]. Proposition 2 (Main). Let X = ( f , g ) : R2 \ D σ → R2 be a differentiable map, where σ > 0 and D σ = { z ∈ R2 :  z  σ }. Suppose that X is Hurwitz: Spc( X ) ⊂ { z ∈ C: (z) < 0}. Then: (i) Any half-Reeb component of either F ( f ) or F ( g ) is a bounded subset of R2 . (ii) There are s  σ and a globally injective local homeomorphism X˜ = ( ˜f , g˜ ) : R2 → R2 such that X˜ and X coincide on R2 \ D s . Moreover, F ( ˜f ) and F ( g˜ ) have no half-Reeb components. Proof. In the proof of (i), we only consider one case. Suppose by contradiction that the foliation, given by the level curves has a unbounded half-Reeb component. By [4, Proposition 1.5], there exists a half-Reeb component of F ( f ) the projection of which is an interval of infinite length. Thus, (a) there are aˆ 0 > σ and A, a half-Reeb component of F ( f ) such that [ˆa0 , +∞) ⊂ Π(A), and the vertical line Π −1 (ˆa0 ) intersects (transversally) both non-compact edges of A. Here Π(x, y ) = x. A half-Reeb component of a fixed foliation contain properly other half-Reeb components, and they are topologically equivalent. In this context, the component is stable under perturbations on their compact face as long as the perturbed arc is also a compact face of some component. Therefore, without lost of generality, we may assume that nearby its endpoints, the compact edge of is made up of arcs of F ( g ). In this way, there exist 0 < a˜ < 12 and an injective, continuous curve γ : (−˜a, 1 + a˜ ) → A such that: − γ ([0, 1]) is a compact edge of A such that both non-compact edges L + γ (0) and L γ (1) are contained in a half-plane {x  aˆ }, for some aˆ > σ . (b.2) The images γ ((−˜a, a˜ )) and γ ((1 − a˜ , 1 + a˜ )) are contained in some leaves of F ( g ) such that sup Π(γ (1 − a˜ , 1 + a˜ )) < inf Π(γ (−˜a, a˜ )). (b.3) For some 0 < δ < 4a˜ there exists an orientation reversing injective function ϕ : [−2δ, 2δ] → (1 − a˜ , 1 + a˜ ) with ϕ (0) = 1 such that f (γ (s)) = f (γ (ϕ (s))). Furthermore, if s ∈ (0, 2δ] then ϕ (s) ∈ (1 − a˜ , 1) and there exists an oriented compact arc of trajectory [γ (s), γ (ϕ (s))] f ⊂ A of F ( f ), connecting γ (s) with γ (ϕ (s)). (b.4) For some 0 < δ < 4a˜ , small enough if s ∈ (0, δ] and γ (s) = (as , d s ) then there exists c s such that q s = (as , c s ) belongs to the open arc (γ (s), γ (ϕ (s))) f ⊂ [γ (s), γ (ϕ (s))] f .

(b.1)

Lemma 4 implies that (c) for every s ∈ (0, δ] as in (b.4), c s < d s . The eigenvalue condition is invariant under addition of constant vectors to maps, therefore we can assume that (d) g (γ (0)) > 0, f = 0 over both non-compact edges of A and f > 0 in the interior of A.

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Fig. 4. Half–Reeb component A, proof of Proposition 2.

Since A is the union of an increasing sequence of compact sets bounded by the compact edge and a compact segment of a leaf. Then, from our selection of the compact edge we have that for every α > Π (γ (0))  inf{as : s ∈ (0, δ]}, large enough there exists sα ∈ (0, δ] as in (b.4) such that the compact arc [γ (sα ), γ (ϕ (sα ))] f projects over (−∞, α ], meeting α (Fig. 4). More precisely,







 

α = sup Π( p ): p ∈ γ (sα ), γ ϕ (sα )

f

.

This defines a closed curve Γα− contained in Π −1 (asα ) ∪ [γ (sα ), γ (ϕ (sα ))] f . If [q sα , p sα ]asα ⊂ Π −1 (asα ) is the vertical segment connecting q sα , of (b.4) with p sα = γ (sα ), then this clock wise oriented curve satisfies

Γα− = [q sα , p sα ]asα ∪ [ p sα , q sα ] f

and

  Π Γα− = [asα , α ],

(5.1)

where the oriented compact arc [ p sα , q sα ] f ⊂ [γ (sα ), γ (ϕ (sα ))] f . We select and fix mα ∈ Γα− ∩ Π −1 (α ), by using Lemma 5 and Lemma 6, respectively we obtain



  X , ∇ f  dt  f ( p sα )

[ p sα ,mα ] f

f x dt + g ( p sα )(α − asα ),

[ p sα ,mα ] f

and



  X , ∇ f  dt  f (q sα )

[mα ,q sα ] f

f x dt + g (mα )(α − asα ).

[mα ,q sα ] f

Therefore, by adding we conclude



  X , ∇ f  dt  f ( p sα )

[ p sα ,q sα ] f

f x dt + g ( p sα )(α − asα )

[ p sα ,q sα ] f

because g in increasing along [ p sα , q sα ] f ⊂ { f = f ( p sα )} and g (mα )  g ( p sα ) = g (γ (0)) > 0.

(5.2)

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(e.1) We claim that, the closed curves Γα− , given in (5.1) define the following functions

α→ 





f dt and α →

 [q sα , p sα ]asα

they are bounded, when thermore,



f x dt

,

[q sα , p sα ] f

α varies in some interval of infinite length contained on [ˆa0 , ∞). Fur lim

α →+∞

f x = 0.

f ( p sα ) [q sα , p sα ] f

In fact, by a perturbation in the compact face if it is necessary, it is not difficult to prove  of F ( f ) such that A  ⊃ A, their boundaries satisfy that there is some half-Reeb component A  ) ⊂  ∂ A \ {compact face} ⊃ ∂ A \ {compact face} and A ⊃ [q sα , p sα ]sα , for all sα . Since the image f (A  f (compact face), the function f is bounded in the closure of A, which contain the compact set  sα [q sα , p sα ]asα . Consequently,



α →





f dt

[q sα , p sα ]asα

is bounded, in some interval of infinite length contained on [ˆa0 , ∞). In order to prove the second part, we apply the Green’s formulae to the map z → (1, 0) in the compact disk D (Γα− ). Since the trace is zero, we obtain





f x dt

= arc length of [q sα , p sα ]asx .

 [q sα , p sα ] f

By compactness we conclude. The last part is directly obtained by using (5.1) and p sα = γ (sα ) because the continuity of the foliation F ( f ) and (d) imply that

lim p sα = γ (0) ∈ { f = 0}.

α →+∞

Therefore (e.1) holds. (e.2) We claim that,



α → +∞ then

if





X , ηαi dt → +∞,

Γα− i where −ηα is an outer normal vector of the close wise oriented curve Γα− .

In the compact arc [ p sα , q sα ] f ⊂ Γα− , the vector that

 Γα−





ηαi is parallel to ∇ f . Thus (5.2) and (e.1) imply

X , ηαi dt  A + g ( p sα )(α − a),

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where the constant A is independent of α and a = min{a s : s ∈ (0, δ]}. Since (d) and (b.2) imply that g ( p sα ) = g (γ (0)) > 0, it is not difficult to obtain (e.2).  ˜ > a such that Γα− satisfies Γ −  X , ηαi  dt > 0. By using the Green’s By (e.2) we select some α α˜

formulae with the map X = ( f , g ), the assumptions over the eigenvalues i.e 0 > Trace( D X z ), imply that

 0>





X , ηαi dt > 0.

Γα˜−

This contradiction concludes the proof of part (i). In order to obtain (ii), we apply Theorem 3 because X satisfies its conditions. This gives the existence of the pair ( X˜ , s) with s  σ and X˜ = ( ˜f , g˜ ) : R2 → R2 a globally injective local homeomorphism such that X˜ and X coincide on R2 \ D s . Furthermore, the last property in (ii) is obtained as a direct application of Corollary 2. Therefore, this proposition holds. 2 Now we prove our main result. 5.1. Proof of Theorem 2 By Proposition 2, there exists a globally injective local homeomorphism X˜ : R2 → R2 such that X˜ and X coincide on some R2 \ D s1 , with s1  σ . In particular, the restriction X | : R2 \ D s1 → R2 is injective. In order to shown the existence of the differentiable extension, consider v = − X˜ (0) joint to the globally injective map X˜ + v. In this context, we can apply the arguments of [15, Theorem 11]. Thus, there are s˜ > s1  σ and a global differentiable vector field Y : R2 → R2 such that: (a.1) R2 \ D s˜  z → Y ( z) = ( X + v )( z) is also injective and Y (0) = 0. (a.2) The map z → Trace( D Y z ) is Lebesgue almost-integrable in whole R2 (see [15, Lemma 7]). (a.3) The index I ( X + v ) is a well-defined number in [−∞, +∞) (see [15, Corollary 13]). X = Y − v is the global Thus, there exist the index of X at infinity, I ( X ) = I ( X + v ). Therefore,  differentiable extension of X | : R2 \ D s˜ → R2 and the pair ( X , s˜ ) satisfies the definition of the index of X at infinity. This concludes the proof of (i). To prove the first part of (ii) we refer the reader to [4, Lemma 3.3]. Furthermore, since Trace( D ( X + v˜ )) = Trace( D X ) < 0, for every constant vector v˜ ∈ R2 we obtain that: (b.1) Given a constant v˜ ∈ R2 , the vector field X + v˜ generates a positive semi-flow on R2 \ D σ . An immediate consequence of (i) is that: if X is Hurwitz, then outside a larger disk both X and Y have no rest points. In addition, by (a.1) the Hurwitz vector field Y has no periodic trajectory γ with D (γ ) contained in R2 \ D σ . As Trace( D Y ) = Trace( D X ) < 0 by Green’s Formulae Y admits at most one periodic trajectory, say γ , such that D (γ ) ⊃ D σ . Consequently: (b.2) There exists s > s˜ such that Y satisfies (a.1), (a.2) and R2 \ D s is free of rest points and periodic trajectories of Y . Under these conditions (b.1) and [15, Theorem 26] imply that: (b.3) For every r  s there exist a closed curve C r transversal to Y contained in the regular set R2 \ D r . In particular, D (C r ) contains D r and C r has transversal contact to each small local integral curve of Y at any p ∈ C r . Moreover, [15, Theorem 28] shown that:

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(b.4) The point at infinity of the Riemann Sphere R2 ∪ {∞} is either an attractor or a repellor of X + v : R2 \ D s → R2 . More specifically, if I ( X ) < 0 (respectively I ( X )  0), then ∞ is a repellor (respectively an attractor) of the vector field X + v. Therefore, (ii) holds and concludes the proof of Theorem 2.

2

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