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Attractors and their invisible parts for skew products with high dimensional fiber. (English) Zbl 1258.37022
Summary: 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 n100m 2 ,m=dim(M), we construct a set n in the space of skew products over the horseshoe with the fiber M having the following properties. Each C 2 -skew product from 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 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 Yu. Ilyashenko and A. Negut [Nonlinearity 23, No. 5, 1199–1219 (2010; Zbl 1204.37017)] to skew products which have an m-dimensional compact manifold as a fiber, m2.
MSC:
37C05Smooth mappings and diffeomorphisms
37C70Attractors and repellers, topological structure