The aim of this paper is to discuss statistical attrac- tors of skew products over the solenoid which have an m-dimensional compact orientable manifold M as a ber and their \"-invisible parts, i.e. a sizable portion of the attractor in which almost all orbits visit it with average frequency no greater than \". We show that there exists a ball D in the space of skew products over a solenoid with the ber M such that each C2-skew product map from D possesses a statistical attractor with an \"-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 dieomorphisms still have attractors with the same properties. Our construction develops the example of [4] to skew products over a solenoid with an m-dimensional ber, m  2. As a consequence, we provide a class of skew product maps acting on S1M such that each map of this class admits a robustly topologically mixing maximal attractor.

Keywords