Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-24T21:00:15.063Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)