Chapter 33 - Inverse limits and dynamical systems

Abstract

This chapter discusses the inverse limits and dynamical systems. If X1,X2,X3, . is a sequence of metric spaces and f1, f2, f3, . is a sequence of mappings, such that fi : Xi+1 →Xi for i = 1, 2, 3, ., by the inverse limit of the inverse limit sequence {Xi, fi} is meant the subset of the product space Πi0Xi that contains the point (x1, x2, x3, .) if and only if fi(xi+1) =xi for each positive integer i. The inverse limit of the inverse limit sequence {Xi, fi} is denoted by lim {Xi, fi}. For convenience of notation, boldface characters are used to denote sequences. The chapter elaborates the concepts related to characterization of chainability, plane embedding, inverse limits on [0, 1], and the property of Kelley. Inverse limits with upper semi-continuous bonding functions and the applications of inverse limits in economics are also discussed in the chapter.