Inverse limits. From continua to chaos
type: Item
instance of: scholarly article
title: Inverse limits. From continua to chaos
zbMATH Open document ID: 1234.54002
DOI: 10.1007/978-1-4614-1797-2
author: W. T. Ingram
author: William S. Mahavier
published in: Developments in Mathematics
publication date: 2011-10-27T00:00:00Z
review text: 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 …
reviewed by: Alejandro Illanes
Mathematics Subject Classification ID: 54-02
Mathematics Subject Classification ID: 54A20
Mathematics Subject Classification ID: 54F15
zbMATH DE Number: 5964727
zbMATH Keywords: atriodicity
zbMATH Keywords: bonding map
zbMATH Keywords: chainability
zbMATH Keywords: fixed point property
zbMATH Keywords: hyperspace
zbMATH Keywords: indecomposability
zbMATH Keywords: interval
zbMATH Keywords: inverse limit
zbMATH Keywords: property of Kelley
zbMATH Keywords: unicoherence
zbMATH Keywords: upper semi-continuous set-valued maps
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4614-1797-2
OpenAlex ID: W4256424670
description: scientific article; zbMATH DE number 5964727