$C^1$-Robust Topologically Mixing Solenoid-Like Attractors and Their Invisible Parts

Abstract

The aim of this paper is to discuss statistical attractors of skew products over the solenoid which have an mdimensional compact orientable manifold M as a fiber and their ε-invisible parts, i.e. a sizable portion of the attractor in which almost all orbits visit it with average frequency no greater than ε. We show that for any n ∈ N large enough, there exists a ball D n in the space of skew products over the solenoid with the fiber M such that each C 2-skew product map from D n possesses a statistical attractor with an ε-invisible part, whose size of invisibility is comparable to that of the whole attractor. Also, it consists of structurally stable skew product maps. In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Our construction develops the example of (Ilyashenko & Negut, 2010) to skew products over the solenoid with an m-dimensional fiber, m ≥ 2. As a consequence, we provide a class of local diffeomorphisms acting on S 1 × M such that each map of this class admits a robustly topologically mixing maximal attractor.