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Singular hyperbolic systems
Author(s):
C.
A.
Morales;
M.
J.
Pacifico;
E.
R.
Pujals
Journal:
Proc. Amer. Math. Soc.
127
(1999),
3393-3401.
MSC (1991):
Primary 58F10, 58F15
Posted:
May 4, 1999
MathSciNet review:
1610761
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Abstract |
References |
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Additional information
Abstract:
We construct a class of vector fields on 3-manifolds containing the hyperbolic ones and the geometric Lorenz attractor. Conversely, we shall prove that nonhyperbolic systems in this class resemble the Lorenz attractor: they have Lorenz-like singularities accumulated by periodic orbits and they cannot be approximated by flows with nonhyperbolic critical elements.
References:
- [1]
- R. Bamón, R. Labarca, R. Mañé, M.J. Pacifico, The explosion of singular cycles, Publ. Math. IHES 78 (1993), 207-232. MR 94m:58152
- [2]
- C. Conley, Isolated sets and the Morse index, CBMS Regional Conf. Series in Math., Amer. Math. Soc. Providence R.I. 38 (1978). MR 80c:58009
- [3]
- W. de Melo, J. Palis, Geometric theory of dynamical systems, Springer-Verlag (1982). MR 84a:58004
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- [5]
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- [6]
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- [7]
- S. Hayashi, Diffeomorphisms in
 satisfy Axiom A, Erg. Th. & Dyn. Sys. 12 (1992), 233-253. MR 94d:58081 - [8]
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- [9]
- R. Labarca, M. J. Pacífico, Stability of singular horseshoe, Topology 25 (1986), 337-352. MR 87h:58106
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 -robust transitive sets for three-dimensional flows, C. R. Acad. Sci. Paris 326 (1998), 81-86. CMP 99:02 - [11]
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Additional Information:
C.
A.
Morales
Affiliation:
Université de Bourgogne, Laboratoire de Topologie, B.P.400, 21011, Dijon Cedex-France
Address at time of publication:
Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C.P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil
Email:
cmorales@u-bourgogne.fr, morales@impa.br
M.
J.
Pacifico
Affiliation:
Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil
Email:
pacifico@impa.br
E.
R.
Pujals
Affiliation:
Instituto de Matemàtica, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, Brazil
Email:
enrique@impa.br
DOI:
10.1090/S0002-9939-99-04936-9
PII:
S 0002-9939(99)04936-9
Keywords:
Lorenz attractor,
hyperbolicity,
Axiom A
Received by editor(s):
November 24, 1997
Received by editor(s) in revised form:
January 22, 1998
Posted:
May 4, 1999
Additional Notes:
This work was partially supported by CNPq-Brasil, Faperj-Brasil, Pronex-Brasil. The first author was partially supported by CNRS-France.
Communicated by:
Mary Rees
Copyright of article:
Copyright
1999,
American Mathematical Society
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