Homoclinic classes for sectional-hyperbolic sets

Abstract

We prove that every sectional-hyperbolic Lyapunov stable set contains a nontrivial homoclinic class.Comment: 5 page

Asymptotically sectional-hyperbolic attractors

“…Another interesting property of sectional-hyperbolic systems is that every Lyapunov stable set with this kind of hyperbolicity has positive topological entropy. This was proved in 2015 by Arbieto, Barragán and Morales by using the Pesin entropy formula for C 1 diffeomorphisms (due to such sets support an SRB-like measure for the time-one map) and the volume expansion of its central subbundle [2].…”

“…Indeed, we construct a compact Riemannian manifold whose boundary is a 4-genus surface and a vector field that is inwardly transverse to the boundary, having an attractor with three singularities, one of them central dissipative. Furthermore, we will use the technique given in [2] to prove that asymptotically sectional-hyperbolic attractors have positive topological entropy under certain hypothesis. More precisely, we assume the existence of a non-atomic SRB-like measure supported in Λ for the time-one map.…”

Sectional-hyperbolic Lyapunov stable sets

“…Let σ be the only Lorenz-like singularity in Λ. By theorem 1.1 in [3], Λ has a nontrivial homoclinic class, and therefore a periodic point q. Since Λ satisfies the property (P ), then there exists σ * ∈ Λ∩Sing(X) such that W u (q)∩W s (σ * ) = ∅, as W u (q) ⊆ Λ for being Λ Lyapunov stable, we have that σ * is Lorenz-like, so σ * = σ, then Λ satisfies the property (S) and by the theorem 2.2, Λ is an attracting.…”

“…The isolated non-trivial hyperbolic sets have periodic orbits by the shadowing lemma [10], but not every sectional-hyperbolic set has periodic orbits as the cherry flow in the torus with a strong contraction (See [4] page 27). It was recently proved that the sectionalhyperbolic Lyapunov stable sets contain a non-trivial homoclinic class [3].…”

“…Let us verify that under the assumptions of the theorem, Λ satisfies the property (S). Let σ the only Lorenz-like singularity in Λ, by theorem 1.1 in [3], Λ has a nontrivial homoclinic class, and therefore a periodic point p. If there exists q ∈ M such that α X (q) = Λ or ω X (q) = Λ, by the corollary 1, every singularity in Λ is Lorenz-like, then σ is the unique singularity of Λ. Also, using the orbit of q we have that p ≺ σ, and since Λ satisfies the conditions of the theorem 10 in [8], so there exists x ∈ Λ such that α X (x) = α X (p) and ω X (x) is a singularity, and this case, necessarily, ω X (x) = {σ}, that is, W u (p) ∩ W s (σ) = ∅.…”