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

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


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