Inverse Limits On Intervals


This chapter provides an introduction to inverse limits for anyone who has completed a basic course 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 introduction culminates with a brief study of inverse limits on [0, 1] with a single bonding map. Much of the remainder of the chapter 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 3 contains an indecomposable continuum. We investigate inverse limits with unimodal bonding maps, logistic bonding maps, tent maps as bonding maps, and certain other families of bonding maps. For the continuum theorists reading this chapter we prove that inverse limits on [0, 1] are characterized by chainability. However, this proof provides insight into the geometric realization of inverse limits as continua and, as such, can increase one's understanding of the variety and complexity of the objects produced by the inverse limit construction. We close the chapter with a proof that the inverse limit of a certain unimodal map is the familiar sin(1/x)-curve. © Springer Science+Business Media, LLC 2012.


Mathematics and Statistics

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Article - Journal

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Publication Date

01 Jan 2012