Ergodic Theory and Dynamical Systems

Équivalence topologique de connexions de selles en dimension 3

a1 Laboratoire de Topologie, UMR 5584 du CNRS, B.P. 47 870, 21078 Dijon Cedex, France (e-mail:
a2 Mathematics Institute, University of Warwick, CV4 7AL Coventry, UK (e-mail:

Article author query
bonatti c   [Google Scholar] 
dufraine e   [Google Scholar] 


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 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 C2 vector field on a 3-manifold, having two hyperbolic zeros p, q of saddle type, such that p admits a contracting complex eigenvalue $-c_p(1 + i\alpha)$, $c_p>0$, $\alpha\neq 0$, and q admits an expanding complex eigenvalue $c_q (1+ i \beta)$, $c_q>0$, $\beta\neq 0$. We assume that a one-dimensional unstable separatrix of p coincides with a one-dimensional 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.

(Received November 13 2002)
(Revised March 17 2003)