Dynamics on spaces of compact subsets with application to brain modeling

[摘要] Let E be an open, bounded subset of R-n and let P(E) be the collection of all subsets of E. The theory of random sets dears with random processes whose outcomes are elements of P(E). Due to the infinite-dimensional nature of P(E), this theory is very technical. In this note we introduce a finite dimensional class of compact subsets of E, K-n(E), which is dense in P(E) yet sufficiently rich for many applications. We study dynamical systems on the space K,(E) by considering transformations r: K-n(E) --> K-n(E) which are constructed from image source data such as occur in the dynamics of the brain. In particular, we establish sufficient conditions for the existence of invariant measures on K-n(E). Under certain conditions these measures are absolutely continuous. We attempt to give meaning to the notion of expansiveness in brain dynamics. (C) 1997 Academic Press.

[发布日期] 1997-12-15