Specification and partial hyperbolicity for flows

Abstract

We prove that if a flow exhibits a partially hyperbolic attractor Λ with splitting T Λ M = E s ⊕ E c and two periodic saddles with different indices such that the stable index of one of them coincides with the dimension of E s then it does not satisfy the specification property. In particular, every singular-hyperbolic attractor with the specification property is hyperbolic. As an application, we prove that no Lorenz attractor satisfies the specification property.1 for t ≥ 0. Now, take a small t 2 > 0 such thatz)) ≤ 2L 0 ε} and t 0 = inf I. By (3.21), (3.25) and (3.26) we have T ∈ I. Assume that t 0 > 0. Sinceby (3.21) and (3.26) we have t 0 − t 2 ∈ I, which is a contradiction. Thus 0 = inf I. ThereforeThe proposition follows taking L = 3L 0 . Lemma 3.10. Let p ∈ Λ be a hyperbolic critical element with dim W ss (p) = dim E s p and U be a subdisk of W u (p) with U ⊂ W u (p). Then there exists µ > 0 such that A(U ) := z∈U F s µ (z) is homeomorphic to U × [−µ, µ] dim E s .

Abundance of Wild Historic Behavior

“…Remark 4.12. Sumi, Varandas and Yamamoto [59] have shown that sectional-hyperbolic attractors (which include singular-hyperbolic attractors as a special 3-dimensional case) do not satisfy specification. Hence, the genericity of wild historic points is more general than the genericity of points with maximal oscillation in systems with specification.…”

Gluing orbit property and partial hyperbolicity

“…The previous result can be understood as an extension of [27,Theorem A] where, analogously, the holonomy map along the strong unstable foliation plays a key role in the proof. In particular, a counterpart of this result for partially hyperbolic flows is expected to hold, in the spirit of [28].…”

“…Related results, on the characterization of the C 1 -interior of diffeomorphisms satisfying any of the previous properties, include [23,24]. However, most dramatically, if one considers the space of C r -diffeomorphisms, r ≥ 1, these two properties seldom occur in the absence of uniform hyperbolicity as proved in [9,27,28].…”

No-shadowing for singular hyperbolic sets with a singularity

“…be the global invariant manifold for x ∈ Λ. Then we have the following lemma which is [13,Lemma 3.7].…”