Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Dynamical properties of singular-hyperbolic attractors

Pages: 67 - 87, Volume 19, Issue 1, September 2007      doi:10.3934/dcds.2007.19.67

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Aubin Arroyo - UNAM - Instituto de Matemáticas, U. Cuernavaca, A.P. 273 Admon. de correos # 3, Cuernavaca, Morelos 62251, México, Mexico (email)
Enrique R. Pujals - IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil (email)

Abstract: We provide a dynamical portrait of singular-hyperbolic transitive attractors of a flow on a 3-manifold. Our Main Theorem establishes the existence of unstable manifolds for a subset of the attractor which is visited infinitely many times by a residual subset. As a consequence, we prove that the set of periodic orbits is dense, that it is the closure of a unique homoclinic class of some periodic orbit, and that there is an SRB-measure supported on the attractor.

Keywords:  Singular-Hyperbolicity, 3-dimensional Flows, Attractors.
Mathematics Subject Classification:  Primary: 37D10, 37D30.

Received: August 2005;      Revised: April 2007;      Available Online: June 2007.