In the present talk we develop an opposite point of view. Namely, we
describe dynamical systems whose attractors have a large part which
is in a sense unobservable. This motivated a notion of $\varepsilon
$-attractor. It is a set (not necessary uniquely defined ) near
which almost all the orbits spend in average more than
$1-\varepsilon $ part of the future time. We discuss the effect of
drastic non-coincidence of actual attractor and $\varepsilon
$-attractor. For $\varepsilon$ sufficiently small, like $10^{-30}$,
the difference between actual attractors and $\varepsilon$
attractors is unobservable in the computer and physical experiments.
Therefore, $\varepsilon$-attractors with small $\varepsilon$ have a
chance to replace actual attractors in applications. Theorems and
conjectures on the subject will be presented.
This is a joint work with Andrei Negut, senior student of Princeton
University.