DYNAMICS ON SPACES OF COMPACT SUBSETS WITH APPLICATION TO BRAIN MODELING

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 proc esses whose outcomes are elements of P(E). Due to the infinite-dimensi onal nature of P(E), this theory is very technical. In this note we in troduce a finite dimensional class of compact subsets of E, K-n(E), wh ich is dense in P(E) yet sufficiently rich for many applications. We s tudy dynamical systems on the space K,(E) by considering transformatio ns r: K-n(E) --> K-n(E) which are constructed from image source data s uch 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 dy namics. (C) 1997 Academic Press.