In this article, we study statistical attractors of skew products which have an m-dimensional compact manifold M as a fiber and their ε-invisible subsets. For any n ≥ 100 m2, m = dim (M), we construct a set [Formula: see text] in the space of skew products over the horseshoe with the fiber M having the following properties. Each C2-skew product from [Formula: see text] possesses a statistical attractor with an ε-invisible part, for an extraordinary value of ε (ε = (m + 1)-n), whose size of invisibility is comparable to that of the whole attractor, and the Lipschitz constants of the map and its inverse are no longer than L. The set [Formula: see text] is a ball of radius O(n-3) in the space of skew products over the horseshoe with the C1-metric. In particular, small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Moreover, for skew products which have an m-sphere as a fiber, it consists of structurally stable skew products. Our construction develops the example of [Ilyashenko & Negut, 2010] to skew products which have an m-dimensional compact manifold as a fiber, m ≥ 2.

We provide an iterated function system on any compact connected m-dimensional manifold with just three diffeomorphisms which are C 1 -robustly minimal. This improves the main result of [Ghane et al., 2010].

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.

Given any compact m-dimensional manifold M , we describe C 1open sets of iterated function systems admitting infinite number of attracting and repelling periodic orbits. We show that attracting periodic orbits are dense in the ambient manifold M . Also, the same property holds for repelling periodic orbits.Moreover, the step skew product maps corresponding to these iterated function systems are topologically transitive.

We consider step and soft skew products over the Bernoulli shift which have an-dimensional closed manifold as a fiber. It is assumed that the fiber maps Hölder continuously depend on a point in the base. We prove that, in the space of skew product maps with this property, there exists an open domain such that maps from this open domain have dense sets of periodic points that are attracting and repelling along the fiber. Moreover, robust properties of invariant sets of diffeomorphisms, including the coexistence of dense sets of periodic points with different indices, are obtained.