Annali di Matematica Pura ed Applicata

, Volume 176, Issue 1, pp 57–72 | Cite as

Global random attractors are uniquely determined by attracting deterministic compact sets

  • Hans Crauel

Abstract

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic.

Mathematics Subject Classification

Primary 58F11 58F12 Secondary 58F39 60D05 60H10 60H15 93E03 

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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1999

Authors and Affiliations

  • Hans Crauel
    • 1
  1. 1.Fachbereich 3 MathematikBerlinFederal Republic of Germany

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