Publications results for "Author=(apaza)"
MR2385698 (2009a:37057)
Apaza, Enoch H.(BR-FRJ-IM); Soares, Regis(BR-FRJ-IM)
Axiom A systems without sinks and sources on $n$-manifolds.
(English summary)
Discrete Contin. Dyn. Syst. 21 (2008), no. 2, 393–401.
37D20 (37C20 57M25)
A dynamical system (diffeomorphism or vector field) defined on a
smooth manifold is said to be "Axiom A'' if its nonwandering set is
hyperbolic and also it is the closure of the set of periodic orbits.
Given a vector field $X$, a "sink'' is a closed hyperbolic
attracting orbit or else a hyperbolic attracting singularity. A
"source'' is a sink for $-X$.
In the mid seventies, R. V. Plykin [Mat. Sb. (N.S.) 94(136) (1974),
243--264, 336; MR0356137 (50 #8608)] proved that, for the
discrete time case,
every Axiom A system on the two-sphere has a sink or a source.
The main result in the paper under review (Theorem 1.1) is that there
exist $C^r$ Axiom A vector fields without sinks and sources on every
compact and boundaryless manifold of dimension $n\geq 3$. The authors
also prove the analogous result for diffeomorphisms. Moreover, they
prove that these examples are robust (Corollary 1 and Corollary 2),
that is, there exists an open set of vector fields in the conditions
of Theorem 1.1.
Finally, the authors prove (Theorem 1.2) that, if every torus
transverse to an Axiom A vector field $X$ on the three-sphere is
unknotted, then $X$ has a sink or a source. Recall that a knot in the
three-sphere that bounds an embedded disk is called "unknotted''.
Reviewed by Mário Bessa
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This list reflects references listed in the original paper as
accurately as possible with no attempt to correct error.
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