Crauel, Hans Random point attractors versus random set attractors. (English) Zbl 1011.37032 J. Lond. Math. Soc., II. Ser. 63, No. 2, 413-427 (2001). Summary: The notion of an attractor for a random dynamical system with respect to a general collection of deterministic sets is introduced. This comprises, in particular, global point attractors and global set attractors. After deriving a necessary and sufficient condition for existence of the corresponding attractors it is proved that a global set attractor always contains all unstable sets of all of its subsets. Then it is shown that in general random point attractors, in contrast to deterministic point attractors, do not support all invariant measures of the system. However, for white noise systems it holds that the minimal point attractor supports all invariant Markov measures of the system. Cited in 63 Documents MSC: 37H99 Random dynamical systems 37C70 Attractors and repellers of smooth dynamical systems and their topological structure 34D45 Attractors of solutions to ordinary differential equations 37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems 93E03 Stochastic systems in control theory (general) Keywords:global set attractor; global point attractor; random dynamical system; deterministic sets; invariant measures; white noise systems; Markov measures PDFBibTeX XMLCite \textit{H. Crauel}, J. Lond. Math. Soc., II. Ser. 63, No. 2, 413--427 (2001; Zbl 1011.37032) Full Text: DOI