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ON THE INJECTIVITY AND ASYMPTOTIC STABILITY AT INFINITY 7 0 ROLANDRABANAL 0 2 Abstract. Let X : U → R2 be a differentiable vector field defined on the n complement of a compact subset of the plane. Let Spc(X) denote the set of u all eigenvalues ofthedifferential DXz,when z variesinwholethe domain U. J We prove that the condition Spc(X) ⊂ {z ∈ C : ℜ(z) < 0} is enough in 4 order toobtain that theinfinity tobeeither anattractor or a repellorforX. 1 Moreprecisely,weshowthat(i)thereexistanunboundedsequenceofcircles, pairwiseboundedanannuluswhoseboundaryistransversaltoXand,(ii)there ] is a neighborhood of infinity free of singularities and periodic trajectories of S X,whereallthetrajectoriesareunbounded. Themainresultisobtainedafter D toprovingtheexistenceofX˜ :R2→R2,atopologicalembeddingwithconvex image, which is equal to X inthe complement of somecompact subset of U. . h Therefore,theassociatedmapofX isinjectiveinaneighborhoodofinfinity. t a m [ 1. Introduction 2 v A very basic example of non–discrete dynamics on the Euclidean n−space is 8 given by a linear vector field on Rn. This linear system has well know properties, 1 forinstanceiftherealpartofitseigenvaluesisnegativethestandardMatrixExpo- 4 nentialFormulagivestheglobalasymptoticstabilityofthe origin. Inthenonlinear 1 case, if Y is a C1−vector field whose singularities have non–degenerate jacobian . 4 matrix, the Hartman–Grobman Theorem [26] tells us that in a sufficiently small 0 neighborhood of the singularity q, the system Y is topologically equivalent to its 7 linear part DY as long as the eigenvalues of DY are not pure imaginary; there- 0 q q : fore, any singular point p where the eigenvalues of DYp have negative real part v is asymptotically stable. This property is closer related with the global asymptotic i X stabilityconjecturewhichclaimsthat: “theknowsingularpointofY willbeglobally r attractor, if all the linear parts of Y are asymptotically stable”. This fact, in the a planar case was obtained in [11, 12, 13] and recently improved in [17]. However, it is already been proved that the global asymptotic stability fails in R3, even for polynomial vector fields [7]. There has been a great interest in the local study of vector fields around their singularities. Asampleofthisstudyistheworkdonein[6,9,30,10,31]. However, in order to understand the global behavior of a planar vector field it is absolutely necessary to study its behavior around to infinity. This is one of the reasons to research the so–called asymptotic stability at infinity [25, 15, 14, 19, 29, 2]. In [15], the authors work with a C1−vector field Y : R2 →R2 for which (i) det(DY ) > 0 z Date:February1,2008. 2000 Mathematics Subject Classification. Primary34D20,34D23;Secondary 26B05,54C45. Key words and phrases. Asymptoticstability,Injectivity, Markus–Yamabeconjecture. Thispaperwaswrittenduringastayoftheauthorati.c.t.p. 1 2 R.RABANAL and (ii) Trace(DY ) < 0 in an neighborhood of infinity. By using a result of [13] z they show that “if such Y has a singularity then, the infinity is either a repellor or an attractor”. Moreover,theauthorsin[15]introducetheIndexI(Y)= Trace(DY) and show that if such Y has a singularity and I(Y) < 0 (resp. I(YR) ≥ 0), then Y is topologically equivalent to z 7→−z that is “the infinity is a repellor”(resp. to z 7→z that is “theinfinity is anattractor”). This Index wasrecentlystudiedin [2], where the authors consider a one-parameter family Y of C1−vector fields; they λ show the bifurcation given by the change in the sign of that Index. Intheimportantwork[24],C.Olechshowedtheexistenceofastrongconnection betweentheasymptoticstabilityofavectorfieldandtheinjectivityofitsassociated map. This connection was strengthened and broadened in subsequent works (see for instance [11]–[20], [25] in the plane, [23, 18] in higherdimensions). The present paperproceedswiththestudyoftheplanarcase,inconnectionwiththebasicspirit in[14], where wasstudied the injectivity atinfinity ofC1−maps Y :R2\D →R2 σ with σ >0 and D ={z ∈C:||z||≤σ}. In this paper [14], the authors provethat σ “except a compact set, the map Y is injective if (1.1) Spc(Y)∩(−ε,+∞)=∅, for some ε>0,” where Spc(Y) denotes the set of all eigenvalues of the differential DX , when z varies in whole the domain R2\D . More precisely, they show the z σ existence of a topological embedding Y˜ : R2 → R2 which is equal to Y in the complement of some disk D with s≥σ. In [16], has been given the differentiable s version of this extension result, that is for differentiable maps not necessarily C1. The author of [29] consider differentiable vector fields X : R2 \D → R2 and σ introduces the so–called B−condition, that is: “there does not exist an unbounded sequence R2 ∋ (x ,y ) → ∞ such that X((x ,y )) → p ∈ R2 and DX has a k k k k (xk,yk) real eigenvalue λ satisfying |x |λ → 0”. In this paper [29] has been proved that k k k in order to obtain the injective extension X˜ of X (like above), it is enough that X verifies the B−condition and Spc(X)∩[0,+∞) = ∅. In the affirmative case, the imageofthe globalmapX˜(R2)is convex. This improves[16]andsothe injectivity result of [14] (see also [20, 8]). In[14],theauthorsalsostudiedC1−vectorfieldsY :R2\D →R2.Byusingthe σ stability resultof[15]they provethat “theinfinitywill be an attractingora repelling singularity of Y, if Spc(Y) does not intersect the union (1.2) (−ε,0]∪{z ∈C:ℜ(z)≥0}, for some ε > 0”, where ℜ(z) denotes the real part of z ∈ C. The differentiable version of this result has been proved in [19]. By using the B−condition, this differentiableresultwasrecentlyimprovedin[29]. Inthepresentarticle,weconsider a differential vector field X :R2\D →R2 whose domain induces a neighborhood σ V :=(R2\D )∪{∞} of∞inthe RiemannsphereR2∪{∞} ina naturalway. We σ provethat the conditionSpc(X)⊂{z ∈C:ℜ(z)<0}is enoughin orderto obtain thatthe vector fieldX :(V,∞)→(R2,0)—which is differentiable in V \{∞},but notnecessarilycontinuousat∞—has∞asanattractingorarepellingsingularity according to the natural definitions below. Throughoutthis article,weshalldenoteby (R2\D )∪{∞}the subspaceofthe σ RiemannsphereR2∪{∞}withtheinducedtopology. Moreover,givenatopological ON THE INJECTIVITY AND ASYMPTOTIC STABILITY AT INFINITY 3 circle C ⊂ R2, the compact disc (resp. open disc) bounded by C, will be denoted by D(C) (resp. D(C)). 2. Statement of the results Given U ⊂ R2 the complement of a compact set. We will consider the differen- tiablemaps(orvectorfields)X :U →R2 whosejacobiandeterminantatanypoint of U is different from zero. 2.1. Injectivity at infinity. Forthesemapsdefinedinaneighborhoodofinfinity, our injectivity result is the following. Theorem 2.1. Let X =(f,g):R2\D →R2 be a differentiable map, where σ >0 σ and D = {z ∈ R2 : ||z|| ≤ σ}. If Spc(X) ⊂ {z ∈ C : ℜ(z) < 0}. Then, there σ are s ≥ σ and a globally injective local homeomorphism X : R2 → R2 with convex image such that X and X coincides on R2\D . s e The map X of Theoreem 2.1 is a differentiable embedding, the image of which may be properly contained in the plane. e By a change in the sign of the map, is not difficult do see that Theorem 2.1 is valid for maps X such that Spc(X) ⊂ {z ∈ C : ℜ(z) > 0}. Furthermore, if A:R2 →R2 is anarbitraryinvertible linear map, then Theorem 2.1 applies to the map A◦X ◦A−1. Theorem 2.1 complements the result of [16], where the authors consider the assumption (1.1), here the negative eigenvalues can approach to zero. Letusproceedtogiveaideaoftheproofofthistheorem. Noticethat,weconsider a differentiable mapX =(f,g):R2\D →R2 whosejacobiandeterminantatany σ point is different from zero. In this context the Local Inverse Function Theorem is true [4, 5]. As a consequence, the level curves {f = constant} make up, on the plane, a C0-foliation F(f). Moreover, the leaves of F(f) are differentiable curves, andtherestrictionoftheotherfunctiongtoanyoftheseleavesisstrictlymonotone; in particular F(f) and F(g) are (topologically) transversal to each other. Like in [17], we orient F(f) (resp. F(g)) in agreement that if L is an oriented leaf of p F(f) (resp. F(g)) thought the point p, then the restriction g| (resp. f| ) is an Lp Lp increasingfunctioninconformitywiththeorientationofL .TheleafL ,sometimes p p will be called trajectory. Remark 2.2. Since,theleavesofthesecontinuousfoliationsaredifferentiablecurves, the following properties holds: (a) Letα:(−a,a)→L beaparametrizationwithα(0)=pofanorientedleaf p of F(f). So,it is not difficult obtainthat dg(α(t))>0.Also,by using the dt local inverse of X at X(p), it is easy to see that for each t ∈(−a,a) there exists η >0 suchthat α′(t)=η X (α(t)) where X (z):=(−f (z),f (z)). t t f f y x (b) Inasimilarway,ifβ :(−a,a)→L (g)isalocalparametrizationofanytra- p jectory ofF(g), foreveryt∈(−a,a),β′(t)=−η X (β(t)) for some η >0. t g t (c) These tangent vector fields X and −X are globally defined but it can be f g discontinuous. e e ThelinearpartsofanymapinTheorem2.1havenopositiveeigenvalues;thus,we saythatthesemapsarefreeofpositiveeigenvalues. Bytheresultsof[13],itisknow that any global map X˜ = (f˜,g˜) : R2 → R2 free of positive eigenvalues is globally 4 R.RABANAL injective, because F(f˜) and F(g˜) does not have any Half–Reeb component (see Definition??). This propertyofthe foliationsis usedinSection??whereweprove thefirstversionofTheorem2.1thatis: “formapsfreeofpositiveeigenvalueswhose foliationshavenounboundedHalf–Reebcomponents”(Theorem??). InSection?? we present some preliminary results about the flux of the vector field X through some leaves of F(f). In Section ??, we include the proof of Theorem 2.1; which proceedby contradiction,we suppose the existence of a Half–Reeb component and by using its boundary we construct a contour whose X−flux is positive. 2.2. Differentiable vector fields. We consider the following system ′ (2.1) z =X(z), where X : R2 \D → R2 is a differentiable vector field for some σ > 0. Since, σ each point on the domain can be an initial condition, such point jointly to (2.1) give an autonomous differential equation, which may have many solutions defined on their maximal intervalof existence. Nevertheless,for every of those trajectories —through the same point, kept fixed— all their local funnel sections are compact connected sets (see [21]); moreover,each trajectory has its two limit sets, α and ω respectively, which are well defined in the sense that only depend of such solution. Noticethat,wecalledtrajectorytothecurvedeterminedbyanysolutiondefinedon its maximalinterval ofexistence. If γ denotes a trajectorythrougha point q ∈U, q then γ+ (resp. γ−) will denote the positive (resp. negative) semi-trajectory of X, q q contained in γ and starting at q. In this way γ =γ−∪γ+ and γ−∩γ+ ={q}. q q q q q q AC0−vectorfieldX :R2\D →R2\{0}(withoutsingularities)canbeextended σ to a map X :((R2\D ∪∞),∞)−→(R2,0) σ (which takes ∞ to 0). In this manner, all questions concerning the local theory of b isolated singularities of planar vector fields can be formulated and studied in the case of the vector field X. For instance, if γ+ (resp. γ−) is an unbounded semi- p p trajectoryofX :R2\D →R2 passingthroughp∈R2\D suchthat, its ω−limit σb σ (resp. α−limit) set is empty, we will also say that γ+ goes to infinity (resp. γ− p p comes from infinity), it will be denoted by ω(γ+)= ∞ (resp. α(γ−) =∞). In this p p context, we may also talk about the phase portrait of X in a neighborhood of ∞. As in [19], we need the following concepts. Definition2.3. Wewillsaythatthe infinity is an attractor (resp. arepellor) for the differentiable vector field X :R2\D →R2 if σ (1) There is a sequence of transversal circles to X tending to infinity, that is for every r ≥ σ there exist a circle C such that D ⊂ D(C ) and C is r r r r transversalto X. (2) ForsomecircleC withs≥σ,allthe trajectoriesγ throughanypointp∈ s p R2\D(C ), satisfy ω(γ+)=∞ that is γ+ go to infinity (resp. α(γ−)=∞ s p p p that is γ− come from infinity). p LetAbeameasurablesubsetofRn,andletF :A→Rbeameasurablefunction. We define as usual F+(z)=max F(z),0 , F−(z)=max −F(z),0 . n o n o ON THE INJECTIVITY AND ASYMPTOTIC STABILITY AT INFINITY 5 Accordingly, we say that F :A→R is Lebesgue almost–integrable if min F+dµ, F−dµ <∞, (cid:26)Z Z (cid:27) A A in which case we define Fdµ= F+dµ− F−dµ, Z Z Z A A A which is a well–defined value of the extended real line [−∞,+∞]. Definition 2.4. LetX :R2\D →R2 be adifferentiable vectorfield. Ifthere are σ s≥σ, and a global differentiable vector field X :R2 →R2 such that: (1) in the complement of the disk D both X and X coincides, and s b (2) the map z 7→Trace(DX ) is Lebesgue almost–integrable in whole R2. z b Then, we define the index of X at infinity I(X) as the number of [−∞,+∞] b given by I(X)= Trace(DX)dx∧dy. ZR2 b This index I(X) is well–defined and it does not depend of the map X, more precisely: b Remark 2.5. If we consider the pairs (s ,X ) and (s ,X ) which satisfy (1) and 1 1 2 2 (2) of Definition 2.4. Thanks to the Green’s formula as presented in [27, 28], and b b to an unbounded sequence of compact discs D(C), with C ⊂R2\D surrounding σ the origin; the proof of Proposition 2.1 in [2] shows that Trace(DX )dx∧dy = Trace(DX )dx∧dy. 1 2 ZR2 ZR2 b b We are now ready to state our result over differentiable vector fields. Theorem 2.6. Let X :R2\D →R2 be a differentiable vector field where σ > 0 σ and D ={z ∈R2 :||z||≤σ}. If Spc(X)⊂{z ∈C:ℜ(z)<0}, then the following σ is satisfied: (1) For all p∈R2\D ,there is a unique positive semi-trajectory starting at p. σ (2) There exist the index of X at infinity, I(X) ∈ [−∞,+∞) such that if I(X)<0 (resp I(X)≥0) the infinity is a repellor (resp. an attractor) of the vector field X+v :R2\D →R2 for some v ∈R2. s This theorem improves the main result of [19], where the authors consider the condition (1.2). In the new assumption, the negative eigenvalues can tend to zero. 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