Modulus of stability for vector fields on 3-manifolds

Abstract. In this paper we give a complete set of invariants (moduli) for mild and strong semilocal equivalence for certain two parameter families of diffeomorphisms on surfaces. These families exhibit a quasi-transversal saddle-connection between a saddle-node and a hyperbolic periodic point. IntroductionGenerically diffeomorphisms with simple recurrence are Morse-Smale and therefore structurally stable [9]. It is conjectured that generically one parameter families of diffeomorphisms, starting at a Morse-Smale one, first bifurcate either by the loss of hyperbolicity of a periodic orbit or by the appearance of a nontransversal saddle-connection [10]. Such one-parameter families may not be stable, but in several relevant cases the classes of equivalence may be parametrized by finitely many real invariants (moduli) as introduced in [8]. To be more precise, let us recall some definitions and results. /. Beloqui and M. J. PacificoA complete characterization of stable generic arcs f^ of diffeomorphisms such that fp is Morse-Smale for /j. <0 and f 0 has simple recurrence was established in [11]. They proved in fact that one of the periodic orbits of/ 0 must be nonhyperbolic and that this orbit is either a saddle-node or a flip. Also, in [11], it is proved that arcs going through either a saddle-node or a flip orbit are strongly stable near these orbits. But arcs passing through a Hopf orbit are not even mildly stable due to the appearance of invariant circles where the restriction of the diffeomorphisms have irrational rotation number. Now, in the same context, if the diffeomorphism at the first point of bifurcation has simple recurrence and all periodic orbits are hyperbolic then their unstable and stable manifolds meet transversally except along one orbit. In this case the arc is not even mildly stable. For instance, let / be a diffeomorphism of a surface which exhibits two hyperbolic fixed points p, q such that W"(p) meets W s (q) along a unique orbit of quasi-transversal intersection (parabolic contact) and let g be near / so that it also exhibits a unique orbit of quasi-transversal intersection between W u (q) and W s (p), where q and p are fixed points for g near q and p. F o r / and g to be equivalent near the closure of this orbit it is necessary and sufficient that log/?"(). For mild equivalence there is one modulus which is A.For one parameter families of vector fields, a complete set of topological invariants near quasi-transversal saddle-connections was established in [14], after the initial contributions of [13] and [1].For generic two-parameter families...

Section: / Beloqui and M J Pacificomentioning

confidence: 99%

mentioning

confidence: 99%

Quasi-transversal saddle-node bifurcation on surfaces

Beloqui

Pacífico

1990

“…To prove the theorem, we consider the 'period 2 (as the Poincare map) orbits'; see figure 1. The theorem of Silnikov implies that there are infinitely many such orbits arbitrarily close to the homoclinic orbit F. We use some knot invariants of the periodic orbits to count the number of twists around the homoclinic orbit F. The ratio of twists in the first and the second turns determines the ratio of eigenvalues A and /JL.…”

Section: Y Togawamentioning

confidence: 99%

“…For vector fields the same question can be studied and in fact for the codimension one case without cycles the existence or not of moduli has been settled through the work in [13], [14], [12] and [1]. However the techniques have to be somewhat different from the ones for diffeomorphisms: the main reason for this is that the topological equivalence does not require to preserve the 'time length'.…”

Section: Y Togawamentioning

confidence: 99%

A modulus of 3-dimensional vector fields

Togawa

1987

“…Sur une surface, il n'y a aucun invariant d'équivalence topologique : deux champs de vecteurs de classe C 1 sur une surface sont toujours topologiquementéquivalents au voisinage d'une connexion de selles. De nombreux auteurs (voir par exemple [2,[8][9][10]) ont donné des invariants d'équivalence topologique de champs de vecteurs au voisinage d'une connexion de codimension 1, en toutes dimensions. En dimension 3, le phénomène que nousétudions est un phénomène de codimension 2 où la situation paraît extrêmement simple : on considère un champ de vecteurs X de R 3 possédant deux zéros hyperboliques p et q de type selle, tels que la variété instable de p et la variété stable de q soient de dimension 1.…”

Section: Introductionunclassified

Équivalence topologique de connexions de selles en dimension 3

Bonatti¹,

Dufraine

2003

Résumé. Nous donnons des invariants complets pour l'équivalence topologique de champs de vecteurs, en dimension 3, au voisinage d'une connexion (par des variétés invariantes de dimension 1) entre des selles possédant des valeurs propres complexes.Abstract. We give a complete set of invariants for the topological equivalence of vector fields on 3-manifolds in the neighborhood of a connection by one-dimensional separatrices between two hyperbolic saddles having complex eigenvalues.More precisely, let X be a C 2 vector field on a 3-manifold, having two hyperbolic zeros p, q of saddle type, such that p admits a contracting complex eigenvalue −c p (1 + iα), c p > 0, α = 0, and q admits an expanding complex eigenvalue c q (1 + iβ), c q > 0, β = 0. We assume that a one-dimensional unstable separatrix of p coincides with a onedimensional stable separatrix of q, and we call the connection the compact segment γ consisting of p, q and their common separatrix (see Figure 1). Such a connection is a codimension two phenomenon.The behaviour of a vector field X in the neighborhood of the connection is given, up to topological conjugacy, by the linear part of X in the neighborhood of p and q and by the transition map between two discs transversal to X in those neighborhoods. First, we choose coordinates in the neighborhoods of p and q in order to put X in canonical form and we show that, up to topological equivalence, we can assume that the transition map is linear. Then our main result is as follows :• When the linear transition map (expressed in the chosen coordinates) is conformal or when its modulus of conformality is small (i.e. less than some function ψ(α, β)), there is no topological invariant : every two such vector fields are topologically equivalent. •On the other hand, when the modulus of conformality is greater than ψ(α, β), there are two topological invariants : one is equal to the ratio α/β, the other one is related to the modulus of conformality of the transition map. Equivalence topologique de connexions de selles 1349Le comportement (àéquivalence topologique près) d'un champ au voisinage d'une connexion est essentiellement décrit par sa partie linéaire au voisinage des deux points selles, et par une transition T , qui est l'holonomie du champ le long de la connexion entre deux disques transverses au champ et contenus dans des ouverts de linéarisation au voisinage des points singuliers.Dans le cas où les valeurs propres associées aux points p et q sont toutes réelles, Vegter (voir [11, 12]) montre que, génériquement, il n'y a pas de module de stabilité : toute perturbation du champ X, suffisamment C 1 -petite et préservant la connexion, est topologiquementéquivalenteà X au voisinage de la connexion. Ce résultat aété généralisé en toutes dimensions par Dias Carneiro et Palis (voir [4]).Nous considérons ici le cas où les points selles p et q possèdent chacun une valeur propre complexe (non-réelle), stable pour p (donc de partie réelle négative) et instable pour q (de partie réelle positive). Nous donnons dans ce c...