A singular-hyperbolic closing Lemma. (English) Zbl 1155.37022
The paper provides a further investigation on the dynamics of singular-hyperbolic sets for flows generated by vector fields on compact connected boundaryless \(3\)-dimensional manifolds. Here, singular-hyperbolic sets are partially hyperbolic sets with hyperbolic singularities and a volume-expanding central subbundle. Two main theorems are shown:
(A) Every recurrent point contained in an attracting singular-hyperbolic set is approximated either by periodic points or by points for which the \(\omega\)-limit set is a singularity.
(B) On every compact \(3\)-manifold there exists a \(C^\infty\)-vector field that exhibits an attracting singular-hyperbolic set with a regular recurrent point that cannot be approximated by periodic points.
(A) Every recurrent point contained in an attracting singular-hyperbolic set is approximated either by periodic points or by points for which the \(\omega\)-limit set is a singularity.
(B) On every compact \(3\)-manifold there exists a \(C^\infty\)-vector field that exhibits an attracting singular-hyperbolic set with a regular recurrent point that cannot be approximated by periodic points.
Reviewer: Christian Pötzsche (München)
MSC:
| 37D30 | Partially hyperbolic systems and dominated splittings |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
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