Non-wandering points outside the attractor in orientation-preserving families

Determine whether there exist parameter regions in orientation-preserving admissible families in which the closure of the unstable manifold of the hyperbolic fixed point z_{+} is a chaotic or strange attractor while the system has non-wandering points outside that closure.

Background

The paper studies planar perturbations of unimodal maps near a homoclinic tangency and focuses on whether chaotic or strange attractors are maximal, meaning that they coincide with the maximal invariant subset of a trapping region. The relevant candidate attractor is the closure of the unstable manifold of the hyperbolic fixed point z_{+}.

For orientation-reversing admissible families, the paper states that the non-wandering set is straightforwardly identified as the union of this closure with the other hyperbolic fixed point. In the orientation-preserving case, however, the authors explain that the problem is only partially solved and known results apply only to selected families. An affirmative answer for Hénon-like maps would be relevant to coexistence phenomena in non-uniformly hyperbolic systems and the Palis program.