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Random Point Attractors Versus Random Set Attractors
Author(s) -
Crauel Hans
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610700001915
Subject(s) - attractor , rössler attractor , mathematics , random compact set , invariant (physics) , markov chain , fixed point , set (abstract data type) , crisis , statistical physics , discrete mathematics , random element , mathematical analysis , random variable , computer science , physics , statistics , nonlinear system , quantum mechanics , bifurcation , mathematical physics , programming language
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.