Conjugacy of normally tangent diffeomorphisms : a tool for treating moduli of stability

Bonckaert, Patrick

Annales de l'Institut Fourier, Volume 40 (1990) no. 1, pp. 213-236.

We give sufficient conditions for the conjugacy of two diffeomorphisms coinciding on a common invariant submanifold V and with equal normal derivative; moreover we obtain that the homeomorphism h realizing this conjugacy satisfies additional inequalities. These inequalities, implying also the existence of the normal derivative of h along V, serve to extend this conjugacy towards regions where moduli of stability are present.

On donne des conditions suffisantes pour que deux difféomorphismes, qui sont égaux sur une même variété invariante $V$ et dont les dérivées dans la direction normale sont aussi égales, soit conjugués ; on obtient en plus que l’homéomorphisme conjuguant $h$ satisfait des inégalités supplémentaires. Ces inégalités, qui impliquent l’existence de la dérivée normale de $h$ le long de $V$, servent à étendre cette conjugaison dans des régions où il y a des modules de stabilité.

[1] M. Berger, P. Gauduchon and E. Mazet, Le Spectre d'une Variété Riemannienne, Lecture Notes in Mathematics 194, Springer-Verlag, Berlin-Heidelberg-New York (1971). | MR | Zbl

[2] P. Bonckaert, F. Dumortier and S. Strien, Singularities of Vector fields on R3 determined by their first nonvanishing jet, Ergodic Theory and Dynamical Systems, 9 (1989) 281-308. | Zbl

[3] H. Brauner, Differentialgeometrie, Viewig, Braunschweig, 1981. | MR | Zbl

[4] M.W. Hirsch, Differential Topology, Springer, New York, 1976. | MR | Zbl

[5] M.W. Hirsch, C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Math., 583, Springer-Verlag, Berlin-Heidelberg-New York, 1977. | MR | Zbl

[6] J.L. Kelley, General Topology, Van Nostrand, New York, 1955. | MR | Zbl

[7] R. Labarca and M.J. Pacifico, Morse-Smale vector fields on 4-manifolds with boundary, Dynamical Systems and bifurcation theory, Pitman series 160, Longman, 1987. | MR | Zbl

[8] S. Lang, Differential Manifolds, Addison-Wesley, Reading Massachusetts, 1972. | MR | Zbl

[9] W. De Melo, Moduli of Stability of two-dimensional diffeomorphisms, Topology 19 (1980), 9-21. | MR | Zbl

[10] W. De Melo and F. Dumortier, A type of moduli for saddle connections of planar diffeomorphisms, J. of Diff. Eq. 75 (1988), 88-102. | MR | Zbl

[11] J. Palis, A differentiable invariant of topological conjugacies and moduli of stability, Astérisque, 51 (1978), 335-346. | Numdam | MR | Zbl

[12] M. Spivak, Differential Geometry, Volume I, Publish or Peris Inc., Berkeley, 1979.

[13] S. Van Strien, Normal hyperbolicity and linearisability, Invent. Math., 87 (1987), 377-384. | MR | Zbl

[14] S. Van Strien, Linearisation along invariant manifolds and determination of degenerate singularies of vector fields in R3, Delft Progr. Rep., 12 (1988), 107-124. | MR | Zbl

[15] S. Van Strien, Smooth linearization of hyperbolic fixed points without resonace conditions, preprint. | Zbl

[16] S. Van Strien and G. Tavares Da Santos, Moduli of stability for germs of homogeneous vector fields on R3, J. of Diff. Eq., 69 (1987), 63-84. | Zbl