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Homoclinic classes for generic \(C^ 1\) vector fields. (English) Zbl 1047.37009
It is known that homoclinic classes for structurally stable Axiom A vector fields are maximal transitive and depend continuously on the periodic orbit data. In addition, if \(H\) is a homoclinic class of \(X,\) then it is saturated. Furthermore, such vector fields do not exhibit cycles formed by homoclinic classes.
The authors establish above mentioned properties for generic \(C^{1}\) vector fields on closed \(n\)-manifolds \(M\) requiring neither structural stability nor Axiom A. They show that homoclinic classes for a residual set of \(C^{1}\) vector fields \(X\) on closed \(n\)-manifolds are maximal transitive and depend continuously on periodic orbit data. Furthermore, \(X\) does not exhibit cycles formed by homoclinic classes. It is also proved that generically a homoclinic class of \(X\) is isolated if and only if it is \(\Omega\)-isolated, and it is the intersection of its stable set with its unstable set. In particular, all the mentioned properties hold for a dense set of \(C^{1}\) vector fields on \(M.\)

MSC:
37C10 Dynamics induced by flows and semiflows
37C20 Generic properties, structural stability of dynamical systems
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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