Inverse Limits on Multi-valued Functions
by
William Mahavier
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 give a definition of continuity, weaker than that of continuity from X into 2^X, using the Hausdorff metric. Then we show that if the maps are continuous then the inverse limit space is compact and connected.

We concentrate on the case where X is the unit interval I=[0, 1], and we have a single bonding map f. By the graph of f we mean the set of all points (x, y) in [0, 1] ×[0, 1] such that y 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. Simple examples show that our definition of continuity is not necessary for the inverse limit to be connected. Not surprisingly, very simple subsets of I ×I yield very complicated inverse limits.

Date received: April 11, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caem-08.