Inverse Limits Of Set Valued Functions
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An Introduction to Inverse Limits with Set-valued Functions
Author: W.T. Ingram
language: en
Publisher: Springer Science & Business Media
Release Date: 2012-08-11
Inverse limits with set-valued functions are quickly becoming a popular topic of research due to their potential applications in dynamical systems and economics. This brief provides a concise introduction dedicated specifically to such inverse limits. The theory is presented along with detailed examples which form the distinguishing feature of this work. The major differences between the theory of inverse limits with mappings and the theory with set-valued functions are featured prominently in this book in a positive light. The reader is assumed to have taken a senior level course in analysis and a basic course in topology. Advanced undergraduate and graduate students, and researchers working in this area will find this brief useful.
Inverse Limits of Set-valued Functions
Much is known about inverse limits of compact spaces with continuous bonding maps. When the requirement that the bonding maps be continuous functions is relaxed, to allow for upper semi-continuous set-valued functions, many of the standard tools used to study inverse limit spaces are no longer available. Also, the structure of the inverse limit becomes more complicated. In this dissertation, the structure of compact subsets of the inverse limit is studied. Necessary and sufficient conditions are given for a compact subset of an inverse limit to be the inverse limit of its projections. Weak crossovers are introduced and used to describe the relationship between a set and the inverse limit of its projections. In the process of examining weak crossovers, tools with potentially broader applications are developed. When set-valued bonding maps are continuous and can be written as a union of continuous functions, sufficient conditions to make the inverse limit a Hausdorff continuum are given. It is also shown that, with another assumption, the inverse limit is in fact decomposable. When the bonding map can be written as a union of set-valued functions between compact subsets of the factor space, a description of the inverse limit is given. This uses a generalization of the itinerary of a point.
Inverse Limits
Author: W.T. Ingram
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-11-06
Inverse limits provide a powerful tool for constructing complicated spaces from simple ones. They also turn the study of a dynamical system consisting of a space and a self-map into a study of a (likely more complicated) space and a self-homeomorphism. In four chapters along with an appendix containing background material the authors develop the theory of inverse limits. The book begins with an introduction through inverse limits on [0,1] before moving to a general treatment of the subject. Special topics in continuum theory complete the book. Although it is not a book on dynamics, the influence of dynamics can be seen throughout; for instance, it includes studies of inverse limits with maps from families of maps that are of interest to dynamicists such as the logistic and the tent families. This book will serve as a useful reference to graduate students and researchers in continuum theory and dynamical systems. Researchers working in applied areas who are discovering inverse limits in their work will also benefit from this book.