The random case of Conley's theorem

Details The random case of Conley's theorem The random case of Conley's theorem

2005-03-26

arxiv.org/abs/math/0503617v2

Nonlinearity 19 (2006), 277-291. http://www.iop.org

Mathematics

Dynamical Systems

15 pages

Scientific paper

10.1088/0951-7715/19/2/002

The well-known Conley's theorem states that the complement of chain recurrent set equals the union of all connecting orbits of the flow $\phi$ on the compact metric space $X$, i.e. $X-\mathcal{CR}(\phi)=\bigcup [B(A)-A]$, where $\mathcal{CR}(\phi)$ denotes the chain recurrent set of $\phi$, $A$ stands for an attractor and $B(A)$ is the basin determined by $A$. In this paper we show that by appropriately selecting the definition of random attractor, in fact we define a random local attractor to be the $\omega$-limit set of some random pre-attractor surrounding it, and by considering appropriate measurability, in fact we also consider the universal $\sigma$-algebra $\mathcal F^u$-measurability besides $\mathcal F$-measurability, we are able to obtain the random case of Conley's theorem.

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