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Attractors and skew products. (English) Zbl 1394.37046

Katok, Anatole (ed.) et al., Modern theory of dynamical systems. A tribute to Dmitry Victorovich Anosov. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2560-9/pbk; 978-1-4704-4119-7/ebook). Contemporary Mathematics 692, 155-175 (2017).
The theory of attractors for dynamical systems is more than half a century old, and during this time many different notions of attractors have been presented, according to the specific long-time behavior to be studied. These distinct definitions were, in general, presented separately from one another, and hence no intrinsic relations among them were to be expected, but this is not always the case. Many of these notions are very closely related, and when put together they open a door to a lot of other properties of the dynamical system in question.
In this paper, the authors present some of these notions and explore their relations. One of these attractors, namely the Milnor attractor, is studied, and its Lyapunov instability is proven as being a topological generic property, for dynamical systems in compact manifolds of dimension greater than one.
They make use of the Gorodetski-Ilyashenko strategy to obtain some robust properties of dynamical systems, using the skew product. Such strategy relies on the following heuristic principle: “all effects observed in a finitely generated free semigroup action by diffeomorphisms of a manifold, may be observed for a single diffeomorphism of a manifold of higher dimension”.
Many results are presented and briefly discussed, such as: skew product over a solenoid and Anosov maps, Hölder skew products and their iterates, perturbations of skew product diffeomorphisms, Hölder property by Gorodetski, generalizations and upper estimates on the Hölder exponent.
Lastly, the authors present a section with results on new robust properties of diffeomorphisms of closed manifolds and manifolds with boundary. For instance, dense orbits with zero intermediate Lyapunov exponents, non-hyperbolic invariant measures, attractors with intermingled basins, local genericity of attractors with intermingled basins, thick attractors, and another developments, as skew products with one-dimensional fibers, invisible attractors, bony attractors and comment on some special ergodic theorems.
For the entire collection see [Zbl 1370.37002].

MSC:

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