Inverse limits with multi valued funtions
by
William S. Mahavier
Department of Mathematics and CS, Emory University

We define the inverse limit of a sequence of functions f₁, f₂, f₃, ... where for each i > 0, f_i maps the compact space X into the space 2^X of compact subsets of X. We assume that each f_n is upper semi continuous and we show that the inverse limit space is compact. We also show that the inverse limit space is connected if each space is connected and f_n(x) is connected for each x and each n, but that this condition is not necessary for connectedness.

For this talk we assume that X=[0, 1] and that we have only one bonding map. By the graph of f we mean the set of all points (x, y) in [0, 1] ×[0, 1] such that y is in f(x). Unlike the case with maps of I into I we find examples where the graph of f is connected but the inverse limit is not connected. We will show a number of simple examples. Not surprisingly, very simple subsets of I ×I can yield very complicated inverse limits.

Date received: February 8, 2001