Inverse Limits on Intervals

Abstract

This chapter provides an introduction to inverse limits for anyone who hascompleted a basiccourse in topology. We begin with some of the fundamental properties of inverse limits on the interval [0, 1] and we include numerous instructive examples. This introductionculminates with a brief study of inverse limits on [0, 1] with a single bonding map. Much of the remainder of thechapter is devoted to inverse limits as they relate to dynamical systems. We begin this with a look at period 3 showing that an inverse limit on [0, 1] with a single bonding map having a periodic point of period 3contains an indecomposablecontinuum. We investigate inverse limits with unimodal bonding maps, logistic bonding maps, tent maps as bonding maps, andcertain other families of bonding maps. For thecontinuum theorists reading thischapter we prove that inverse limits on [0, 1] are characterized bychainability. However, this proof provides insight into the geometric realization of inverse limits ascontinua and, as such,can increase one’s understanding of the variety andcomplexity of the objects produced by the inverse limitconstruction. Weclose thechapter with a proof that the inverse limit of acertain unimodal map is the familiar sin(1/x)-curve.