Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Partially hyperbolic fixed
points with constraints


Author: Patrick Bonckaert
Journal: Trans. Amer. Math. Soc. 348 (1996), 997-1011
MSC (1991): Primary 58F14, 58F36, 34C20
DOI: https://doi.org/10.1090/S0002-9947-96-01469-9
MathSciNet review: 1321568
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the local conjugacy, at a partially hyperbolic fixed point, of a diffeomorphism (vector field) to its normally linear part in the presence of constraints, where the change of variables also must satisfy the constraints. The main result is applied to vector fields respecting a singular foliation, encountered, by F. Dumortier and R. Roussarie, in the desingularization of families of vector fields.


References [Enhancements On Off] (What's this?)

  • [BP] Beloqui, J. and Pacifico, M.J., Quasi-transversal saddle-node bifurcation on surfaces, Ergod. Th. & Dynam. Sys. 10 (1990), 63--88. MR 92a:58104
  • [Bo] Bonckaert, P., On the continuous dependence of the smooth change of coordinates in parametrized normal form theorems, Journal of Differential Equations 106 (1993), 107--120. MR 94j:58150
  • [Br] Broer, H., Formal normal form theorems for vector fields and some consequences in the volume preserving case, Dynamical systems and Turbulence, Warwick, Lecture Notes in Math. 898, Springer-Verlag, Berlin-Heidelberg-New York, 1981. MR 83j:58085
  • [CP] Carneiro, M. and Palis, J., Bifurcations and global stability of families of gradients, Publ. Math. IHES 70 (1989), 103--168. MR 91f:58048
  • [DRR] Dumortier, F. D., Rodrigues, P. and Roussarie, R., Germs of diffeomorphisms in the plane, Lecture Notes in Mathematics 902, Springer-Verlag, 1981. MR 83f:58008
  • [DR1] Dumortier, F. and Roussarie. R., Germes de difféomorphismes et de champs de vecteurs en classe de différentiabilité finie, Ann. Inst. Fourier 33 (1983), 195--267. MR 84m:58114
  • [DR2] Dumortier, F. and Roussarie, R., Canard cycles and center manifolds, Memoirs of the AMS (to appear).
  • [IY] Il$'$yashenko, Y. and Yakovenko, S., Finitely smooth normal forms of local families of diffeomorphisms and vector fields, Russian Math. Surveys 46:1 (1991), 1--43. MR 92i:58165
  • [M] Mather, J., Stability of $C^\infty $ mappings: II Infinitesimal stability implies stability, Ann. of Math. 89 (1969), 254--291. MR 41:4582
  • [NPT] Newhouse, S., Palis, J. and Takens, F., Bifurcation and stability of families of diffeomorphisms, Publ. Math. IHES 57 (1983), 5--71. MR 84g:58080
  • [PT] Palis, J. and Takens, F., Stability of parametrized families of gradient vector fields, Annals of Math. 118 (1983), 383--42. MR 85i:58093
  • [R] Roussarie, R., Modèles locaux de champs et de formes, Astérisque 30 (1975), 1--181. MR 55:13444
  • [T] Takens, F., Partially hyperbolic fixed points, Topology 10 (1971), 133--147. MR 46:6399

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F14, 58F36, 34C20

Retrieve articles in all journals with MSC (1991): 58F14, 58F36, 34C20


Additional Information

Patrick Bonckaert
Affiliation: Limburgs Universitair Centrum, B-3590 Diepenbeek, Belgium
Email: pbonckae@luc.ac.be

DOI: https://doi.org/10.1090/S0002-9947-96-01469-9
Keywords: Partially hyperbolic fixed point, normal form, constraint, structure preserving
Received by editor(s): March 8, 1994
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society