This paper deals with attractors of generic dynamical systems. We introduce the notion of ε-invisible set, which is an open set of the phase space in which almost all orbits spend on average a fraction of time no greater than ε. For extraordinarily small values of ε (say, smaller than 2 ~(-100), these are large neighbourhoods of some parts of the attractors in the phase space which an observer virtually never sees when following a generic orbit. For any n ≥ 100, we construct a set Q_n in the space of skew products over a solenoid with the fibre a circle having the following properties. Any map from Q_n is a structurally stable diffeomorphism; the Lipschitz constants of the map and its inverse are no greater than L (where L is a universal constant that does not depend on n, say L < 100). Moreover, any map from Q_n has a 2~(-n)-invisible part of its attractor, whose size is comparable to that of the whole attractor. The set Q_n is a ball of radius O(n~(-2) in the space of skew products with the C~1 metric. It consists of structurally stable skew products. Small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and is practically never seen by the observer.
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