Vol. 8, No. 2, 2015

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On attractors and their basins

Alexander Arbieto and Davi Obata

Vol. 8 (2015), No. 2, 195–209
Abstract

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
Mathematical Subject Classification 2010
Primary: 37C10, 37C20
Milestones
Received: 6 December 2011
Revised: 12 December 2013
Accepted: 8 January 2014
Published: 3 March 2015

Communicated by Kenneth S. Berenhaut
Authors
Alexander Arbieto
Instituto de Matemática
Universidade Federal do Rio de Janeiro
P. O. Box 68530
21945-970 Rio de Janeiro
Brazil
Davi Obata
Instituto de Matemática
Universidade Federal do Rio de Janeiro
P. O. Box 68530
21945-970 Rio de Janeiro
Brazil