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A dichotomy for three-dimensional vector fields

Published online by Cambridge University Press:  23 September 2003

C. A MORALES  and
M. J PACIFICO
C. A MORALES
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro C. P. 68.530, CEP 21.945-970, Rio de Janeiro, R. J., Brazil (e-mail: [email protected], [email protected])
M. J PACIFICO
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro C. P. 68.530, CEP 21.945-970, Rio de Janeiro, R. J., Brazil (e-mail: [email protected], [email protected])
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Abstract

We prove that a generic C1 vector field on a closed 3-manifold either has infinitely many sinks or sources or else is singular Axiom A without cycles. Singular Axiom A means that the non-wandering set of the vector field has a decomposition into compact invariant sets, each being either a hyperbolic basic set or a singular hyperbolic attractor (like the Lorenz-like ones) or a singular hyperbolic repeller. An attractor is a transitive set which attracts all nearby future orbits, and a repeller is an attractor for the time-reversed flow. Our result implies that generic C1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 23 , Issue 5 , October 2003 , pp. 1575 - 1600
Copyright
2003 Cambridge University Press

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