Attractors and their Invisible Parts for skew Products with High Dimensional Fiber

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 ≥ 100m, m = dim(M), we construct a set Rn in the space of skew products over the horseshoe with the fiber M having the following properties. Each C-skew product from Rn 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 Rn is a ball of radius O(n−3) in the space of skew products over the horseshoe with the C-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 anm-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.