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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 \(\epsilon\)-invisible subsets. For any \(n \geq 100 m^2, m= \dim(M)\), 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 possesses a statistical attractor with an \(\epsilon\)-invisible part, for an extraordinary value of \(\epsilon(\epsilon = (m + 1)^{-n})\), 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 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, \(m\geq 2\).

MSC:

37C05 Dynamical systems involving smooth mappings and diffeomorphisms
37C70 Attractors and repellers of smooth dynamical systems and their topological structure

Citations:

Zbl 1204.37017
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References:

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