Dynamics on Spaces of Compact Subsets with Application to Brain Modeling*

Let E be an open, bounded subset of Rn and let P E. be the collection of all subsets of E. The theory of random sets deals 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,Kn E., which is dense in P E.yet sufficiently rich for many applications. We study dynamical systems on the space Kn E. by considering transformations t: Kn E.ªKn 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 Kn E.. Under certain conditions these measures are absolutely continuous. We attempt to give meaning to the notion of expansiveness in brain dynamics