June  2008, 21(2): 393-401. doi: 10.3934/dcds.2008.21.393

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


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

Received  May 2007 Revised  November 2007 Published  March 2008

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.
Citation: Enoch H. Apaza, Regis Soares. Axiom a systems without sinks and sources on $n$-manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 393-401. doi: 10.3934/dcds.2008.21.393

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