Ergodic Theory and Dynamical Systems



Homoclinic classes for generic C^1 vector fields


C. M. CARBALLO a1, C. A. MORALES a2 and M. J. PACIFICO a2
a1 Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente, 225 CEP 22453-900, Rio de Janeiro, RJ, Brazil (e-mail: [email protected])
a2 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])

Article author query
carballo cm   [Google Scholar] 
morales ca   [Google Scholar] 
pacifico mj   [Google Scholar] 
 

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.

(Received June 6 2000)
(Revised May 7 2002)