  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.
