Dynamical Properties of Singular Hyperbolic Attractors

Enrique Pujals | Arroyo, Aubin

Keywords: Singular Hyperbolic Attractor Invariant Manifolds Flows equilibrium

Singular hyperbolicity is a weaker form of hyperbolicity that is found on any $C^1$-robust transitive set with singularities of a flow on a three-manifold, like the Lorenz Attractor, \cite{MPP0}. In this work we are concerned in the dynamical properties of such invariant sets. For instance, we obtain that if the attractor is singular hyperbolic and transitive, the set of periodic orbits is dense. Also we prove that it is the closure of a unique homoclinic class of some periodic orbit. A corollary of the first property is the existence of an SRB measure supported on the attractor. These properties are consequences of a theorem of existence of unstable manifolds for transitive singular hyperbolic attractors, not for the whole set but for a subset which is visited infinitely many times by a resiudal subset of the attractor. Here we give a complete proof of this theorem, in a slightly more general context. A consequence of these techniques is that they provide a sufficient condition for the $C^1$-robust transitivity.

Anexos:

• dynamicalproperties.pdf