Inverse limits. From continua to chaos
Inverse limits are an important tool to study some aspects of the theory of continua. They provide a powerful tool for constructing interesting and complicated examples. This book is an excellent introduction to this topic and it is also useful for researchers since it includes the new developments in this area. The spaces considered here are always continua. In Chapter 1, the authors develop a detailed theory of inverse limits on the interval \([0,1]\). They include basic theory, many examples and the study of inverse limits for some particular families of maps. Some of the examples treated in this chapter are of interest to dynamicists such as the logistic and the tent families. This chapter provides an introduction to inverse limits that can be read for anyone who has completed a basic course in topology. Chapter 2 presents the theory of inverse limits with bonding functions being upper semi-continuous set-valued maps, a generalization that has been recently introduced by the authors and has become an important topic of research. The authors include a discussion of the main results that have been obtained in this area. They discuss the properties which hold in the classical theory of inverse limits and that are not valid for inverse limits with upper semi-continuous set-valued bonding maps. They present numerous examples with useful figures to show how complicated an inverse limit can be, even when a single upper semi-continuous set-valued function from \([0,1]\) into \([0,1]\) is taken. In Chapter 3, a discussion is included of how some properties of the factor spaces and/or the maps give properties of the inverse limit. In this direction the following properties are considered: indecomposability, atriodicity, unicoherence, irreducibility, chainability, property of Kelley and fixed point property. Chapter 4 is devoted to presenting Morton Brown's theorem that allows certain adjustments in the sequence of bonding maps without changing the topology on the inverse limit. Chapter 5 contains background material on the Hilbert cube. The book finishes with a very complete bibliography that includes 591 items.
scientific article; zbMATH DE number 5964727