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Homoclinic classes for generic C^1 vector fields

Published online by Cambridge University Press:  22 September 2003

C. M. CARBALLO ,
C. A. MORALES  and
M. J. PACIFICO
C. M. CARBALLO
Affiliation:
Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente, 225 CEP 22453-900, Rio de Janeiro, RJ, Brazil (e-mail: [email protected])
C. A. MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil (e-mail: [email protected], [email protected])
M. J. PACIFICO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro, CP 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil (e-mail: [email protected], [email protected])
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Abstract

We prove that homoclinic classes for a residual set of C^1 vector fields X on closed n-manifolds are maximal transitive, and depend continuously on periodic orbit data. In addition, X does not exhibit cycles formed by homoclinic classes. We also prove that a homoclinic class of X is isolated if and only if it is \Omega-isolated, and it is the intersection of its stable set with its unstable set. All these properties are well known for structurally stable Axiom A vector fields.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 2 , April 2003 , pp. 403 - 415
Copyright
2003 Cambridge University Press

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