Attractors and their invisible parts for skew products with high dimensional fiber

摘要

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 m ~2, m = dim(M), we construct a set $mathcal{R}-n$ in the space of skew products over the horseshoe with the fiber M having the following properties. Each C ~2-skew product from $mathcal{R}-n$ 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 $mathcal{R}-n$ is a ball of radius O(n ~(-3)) in the space of skew products over the horseshoe with the C ~1-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.