Mapping Theorems

Abstract

In this chapter, we include some theorems on mappings of inverse limit spaces. Although the subsequence theorem for inverse limits with mappings does not hold in general for inverse limits with set-valued functions, there is a version for upper semicontinuous functions that gives a mapping between inverse limits including, specifically, a mapping of ${{\lim }\atop{\longleftarrow}} \mathbf{f}$ onto ${{\lim }\atop{\longleftarrow}} \mathbf{f}^{2}$ for inverse limits with a single bonding function. The shift homeomorphisms between inverse limits with mappings also do not carry over as homeomorphisms to the set-valued case. Instead, one shift is a mapping and the other is a set-valued function. A generalized conjugacy theorem rounds out this chapter.