A spectral decomposition for singular-hyperbolic sets

Abstract

We extend the Spectral Decomposition Theorem for hyperbolic sets to singular-hyperbolic sets on 3-manifolds. We prove that an attracting singularhyperbolic set with dense periodic orbits and a unique equilibrium of a C r vector field, where r ≥ 1, is a finite union of transitive sets; the union is disjoint or the set contains finitely many distinct homoclinic classes. If the vector field is C r -generic, the union is in fact disjoint.

Intransitive Sectional-Anosov Flows on 3-manifolds

“…An example with a unique singularity was given in [6], and in [19] was proved that every Venice mask X (1) with one equilibrium satisfies the following properties:…”

“…Particularly, was proved in [19], [18] that every Venice mask with a unique singularity has these properties. The above observations motivate the following questions, 1.…”

“…So, a sectional-Anosov flow is said a Venice mask if it has dense periodic orbits but is not transitive. An example of Venice mask with a unique singularity was given in [6], and for three singularities was provided in [19]. Recently, [13] showed the construction the examples with two equilibria.…”

“…Recently, [13] showed the construction the examples with two equilibria. They are characterized because the maximal invariant set is non disjoint union of two homoclinic classes [19], [18], [6], and the intersection between these classes is contained in the closure of the union of unstable manifolds of the singularities.…”

SURVEY Towards a global view of dynamical systems, for the C1-topology

“…A large sequence of papers by Morales, Pacífico and Pujals defines the notion of singular hyperbolic sets and proves that robustly isolated singular classes of three-dimensional flows are singular hyperbolic (see for instance [MP2,MPP]). A large sequence of papers by Morales, Pacífico and Pujals defines the notion of singular hyperbolic sets and proves that robustly isolated singular classes of three-dimensional flows are singular hyperbolic (see for instance [MP2,MPP]).…”