Abstract
A transitive set Λ of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC 1 generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC 1 vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].
Similar content being viewed by others
References
M. C. Arnaud,Création de connexions en topologie C 1. C. R. Acad. Sci. Paris, Série I,329 (1999), 211–214.
N.P. Bhatia and G.P. Szego,Stability theory of dynamical systems. Springer-Verlag, Heidelberg, 1979.
C. Bonatti, A. Pumariño and M. Viana,Lorenz attractors with arbitrary expanding dimension. C. R. Acad. Sci. Paris,325 (1997), Série I: 883–888.
C. M. Carballo, C. A. Morales and M. J. Pacifico,Homoclinic classes for generic C 1 vector fields. Preprint MAT.07/2000 PUC-Rio, 2000.
W. de Melo and J. Palis,Geometric Theory of Dynamical Systems-An Introduction. Springer Verlag, Berlin, 1982.
S. Hayashi,Connecting invariant manifolds and the solution of the C 1 stability and Μ-stability conjectures for flows. Annals of Math.,145 (1997), 81–137.
S. Hayashi,Hyperbolicity, stability, and the creation of homoclinic points. In Documenta Mathematica, Extra Volume ICM, Vol. II, 1998, 1998.
M. Hirsch, C. Pugh and M. Shub,Invariant manifolds, volume 583 of Lect. Notes in Math. Springer Verlag, Berlin, 1977.
M. Hurley,Attractors: persistence, and density of their basins. Trans. AMS,269 (1982), 247–271.
K. Kuratowski,Topology II. Academic Press- PWN-Polish Sci. Publishers Warszawa, 1968.
C. Morales and M. J. Pacifico, Non-transitive attracting sets for 3-flows. In preparation.
C. Morales and M. J. Pacifico, Lyapunov stability of generic ω-limit sets. Preprint, 2000.
C. Morales, M. J. Pacifico and E. Pujals,On C 1 robust singular transitive sets for three-dimensional flows. C. R. Acad. Sci. Paris,326 (1998), Série I: 81–86.
C. Morales and E. Pujals,Singular strange attractors on the boundary of Morse-Smale systems. Ann. Sci. École Norm. Sup.,30 (1997), 693–717.
D. V. Turaev and L. P. Shilnikov,An example of a wild strange attractor. Mat. Sbornik,189 (1998), 137–160.
Author information
Authors and Affiliations
Additional information
This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq.
About this article
Cite this article
Carballo, C.M., Morales, C.A. & Pacifico, M.J. Maximal transitive sets with singularities for genericC 1 vector fields. Bol. Soc. Bras. Mat 31, 287–303 (2000). https://doi.org/10.1007/BF01241631
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01241631