Minimal forward random point attractors need not exist

Abstract

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [1] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward point attractors but no minimal one remained open. Here we provide such an example.

Asymptotics for a class of iterated random cubic operators

“…It is shown in [4] that minimal point attractors in the weak or pullback sense always exist. For forward point attractors this need not be the case (see [16]) but in the particular set-up of the following theorem we will see that a minimal forward point attractor exists.…”