## 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

Zbl 1204.37017
Full Text:

### References:

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