Maximal transitive sets with singularities for genericC
1 vector fields |
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Authors: | C M Carballo C A Morales M J Pacifico |
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Institution: | (1) Departamento de Matemática, PUC-Rio, Rio de Janeiro, RJ, Brazil;(2) Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970 Rio de Janeiro, RJ, Brazil |
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Abstract: | A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC
1 generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC
1 vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in 3] and 15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq. |
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Keywords: | transitive sets maximal transitive sets transitive sets with singularities |
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