Axiom a systems without sinks and sources on $n$-manifolds

Enoch H. Apaza; Regis Soares

doi:10.3934/dcds.2008.21.393

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2008, Volume 21, Issue 2: 393-401. Doi: 10.3934/dcds.2008.21.393

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Axiom a systems without sinks and sources on $n$-manifolds

Enoch H. Apaza^1, and
Regis Soares^1,

1.
Instituto de Matematica, Universidade Federal do Rio de Janeiro, P.O. Box 68530, Rio de Janeiro, RJ 21945-970

Received: May 2007

Revised: November 2007

Published: May 2008

Abstract / Introduction Related Papers Cited by

Abstract

It is well known that every Axiom A diffeomorphism defined in the 2-sphere $S^{2}$ has a sink or a source [19]. A natural question is if this property is still true for higher dimensional Axiom A diffeomorphisms and Axiom A vector fields. In this paper we give a negative answer to this question: we prove that for every closed manifold of dimension $n\geq 3$ there are a $C^1$ open set of Axiom A diffeomorphisms and a $C^1$ open set of Axiom A vector fields without sinks and sources. We also show that a sufficient condition for an Axiom A vector field in $S^3$ to exhibit a sink or a source is that every torus in $S^3$ transverse to $X$ is unknotted.

Keywords:

Axiom A,
sink,
source,
knot.

Mathematics Subject Classification: Primary: 37D20; Secondary: 57M25.

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