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On attractors and
their basins
Alexander Arbieto and Davi Obata
Vol. 8 (2015), No. 2, 195–209
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Abstract |
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We prove that the map assigning to a given vector field the Lebesgue measure of the union of the basins of its attractors is lower semicontinuous in a residual subset of vector fields. Moreover, we prove that the Lebesgue measure of the union of the basins of attractors of a generic sectional axiom A vector field is total. For this, we also improve a result of Morales about sectional-hyperbolic sets. We also remark that homoclinic classes are topologically ergodic and that for a generic tame diffeomorphism, the union of the stable manifolds of the hyperbolic periodic orbits is dense in the manifold. |
Keywords
attractors, basin, sectional axiom A, sectional hyperbolic,
basin of attraction
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Mathematical Subject Classification 2010
Primary: 37C10, 37C20
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Milestones
Received: 6 December 2011
Revised: 12 December 2013
Accepted: 8 January 2014
Published: 3 March 2015
Communicated by Kenneth S. Berenhaut
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Authors |
| Alexander Arbieto | |
| Instituto de Matemática Universidade Federal do Rio de Janeiro P. O. Box 68530 21945-970 Rio de Janeiro Brazil |
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| Davi Obata | |
| Instituto de Matemática Universidade Federal do Rio de Janeiro P. O. Box 68530 21945-970 Rio de Janeiro Brazil |
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