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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 $n\ge 100{m}^{2},m=dim\left(M\right)$, we construct a set ${ℛ}_{n}$ in the space of skew products over the horseshoe with the fiber $M$ having the following properties. Each ${C}^{2}$-skew product from possesses a statistical attractor with an $ϵ$-invisible part, for an extraordinary value of $ϵ\left(ϵ={\left(m+1\right)}^{-n}\right)$, 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 ${ℛ}_{n}$ is a ball of radius $O\left({n}^{-3}\right)$ 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, $m\ge 2$.
##### MSC:
 37C05 Smooth mappings and diffeomorphisms 37C70 Attractors and repellers, topological structure