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Équivalence topologique de connexions de selles en dimension 3
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Bonatti, C. (Christian) and Dufraine, Emmanuel. (2003) Équivalence topologique de connexions de selles en dimension 3. Ergodic Theory and Dynamical Systems, Vol.23 (No.5). pp. 13471381. ISSN 01433857

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Official URL: http://dx.doi.org/10.1017/S0143385703000130
Abstract
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.
We give a complete set of invariants for the topological equivalence of vector fields on 3manifolds in the neighborhood of a connection by onedimensional separatrices between two hyperbolic saddles having complex eigenvalues.
More precisely, let X be a C2 vector field on a 3manifold, having two hyperbolic zeros p, q of saddle type, such that p admits a contracting complex eigenvalue $c_p(1 + i\alpha)$, $c_p>0$, $\alpha\neq 0$, and q admits an expanding complex eigenvalue $c_q (1+ i \beta)$, $c_q>0$, $\beta\neq 0$. We assume that a onedimensional unstable separatrix of p coincides with a onedimensional stable separatrix of q, and we call the connection the compact segment $\gamma$ 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 $\psi(\alpha,\beta)$), 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 $\psi(\alpha,\beta)$, there are two topological invariants: one is equal to the ratio $\alpha/\beta$, the other one is related to the modulus of conformality of the transition map.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Manifolds (Mathematics), Eigenvalues, Mappings (Mathematics), Topology, Geometry, Differential 
Journal or Publication Title:  Ergodic Theory and Dynamical Systems 
Publisher:  Cambridge University Press 
ISSN:  01433857 
Official Date:  October 2003 
Volume:  Vol.23 
Number:  No.5 
Page Range:  pp. 13471381 
Identification Number:  10.1017/S0143385703000130 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
Description:  Text in French with a summary in English. 
References:  [1] G. R. Belitsky. Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Funct. Anal. Appl. 7, (1973), 268–277. 
URI:  http://wrap.warwick.ac.uk/id/eprint/774 
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