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.
Highlights
Hyperbolicity is the paradigm of stability for diffeomorphisms and flows without singularities; where some conditions on the behavior of the derivative constrain the dynamic to a robust scenario. They are so called Axiom A or Uniform Hyperbolic systems and they are well understood after the work of Smale [Sm] and others. Such systems are structurally stable, the set of periodic orbits is dense in the non-wandering set, and they have a spectral decomposition on a finite union of homoclinic classes
When we focus on flows defined on a closed manifold M the accumulation of regular orbits on fixed points rule out such hyperbolic structure; it may produce a new phenomena which in some cases behaves very like the uniform hyperbolic systems
A striking example of this is the Lorenz Attractor [Lo]: an invariant set of a flow given by the solutions of the following polynomial vector field in R3
Summary
Hyperbolicity is the paradigm of stability for diffeomorphisms and flows without singularities; where some conditions on the behavior of the derivative constrain the dynamic to a robust scenario. In [Co] he prove the following: if Λ is a singular hyperbolic transitive attractor of a C1+α vector field, α > 0, with dense periodic orbits it has an SRB measure. A deeper consequence of the theorem of existence of unstable manifolds is that it allows us to find a sufficient condition to obtain C1-robustly transitive sets with singularities. Such condition is (H*), below, which roughly speaking states that there are some neighborhoods of Sing(Λ) where the maximal invariant set of the complement in U of such neighborhoods is a basic piece.
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