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arXiv · 0901.0316

Invisible Parts of Attractors

Yulij Ilyashenko·Andrei Negut
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Abstract

This paper deals with the attractors of generic dynamical systems. We introduce the notion of epsilon-invisible set, which is an open set in which almost all orbits spend on average a fraction of time no greater than epsilon. For extraordinarily small values of epsilon (say, smaller than 2^{-100}), these are areas of the phase space which an observer virtually never sees when following a generic orbit. We construct an open set in the space of all dynamical systems which have an epsilon-invisible set that includes parts of attractors of size comparable to the entire attractor of the system, for extraordinarily small values of epsilon. The open set consists of C^1 perturbations of a particular skew product over the Smale-Williams solenoid. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and practically never seen by the observer.

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Yulij Ilyashenko, Andrei Negut. 2009-01-03. Invisible Parts of Attractors. https://doi.org/10.1088/0951-7715%2F23%2F5%2F010

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