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Über dieses Buch

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. ​



Chapter 1. Basics

The study of inverse limits with set-valued functions was introduced in 2004 and has developed into a rich topic for research in topology. One path into this subject can be found by working through examples of such inverse limits. A natural starting point for such an undertaking is consideration of examples with a single bonding function on [0,1] having closed set values. In this chapter we include the basic definitions and theorems needed to read the remainder of the book much of which is driven by examples. We state and prove our theorems on [0,1]; with minor modifications the proofs generally are valid in a much more general setting such as compact metric spaces or even compact Hausdorff spaces.
W. T. Ingram

Chapter 2. Connectedness

A fundamental question about inverse limits with set-valued bonding functions relates to the connectedness of the inverse limit. For inverse limits on compact, connected factor spaces with bonding functions that are mappings, the inverse limit is always connected. However, for inverse limits with set-valued functions as bonding functions, the inverse limit is rarely connected. One might suspect that this is due to the fact that the graph of an upper semicontinuous function on a compact, connected space can fail to be connected, but the reasons go much deeper. In this chapter we study connectedness of inverse limits on [0,1] with set-valued functions.
W. T. Ingram

Chapter 3. Mappings versus Set-Valued Functions

Inverse limits with upper semicontinuous bonding functions exhibit fundamental differences from inverse limits with mappings in the sense that the theorems that hold when the bonding functions in an inverse limit sequence are mappings almost always fail if the bonding functions are set-valued. This chapter is devoted to examining some of those differences. Of course, these differences provide a source for research questions.
W. T. Ingram

Chapter 4. Mapping Theorems

In this chapter, we include some theorems on mappings of inverse limit spaces. Although the subsequence theorem for inverse limits with mappings does not hold in general for inverse limits with set-valued functions, there is a version for upper semicontinuous functions that gives a mapping between inverse limits including, specifically, a mapping of \({{\lim }\atop{\longleftarrow}} \mathbf{f}\) onto \({{\lim }\atop{\longleftarrow}} \mathbf{f}^{2}\) for inverse limits with a single bonding function. The shift homeomorphisms between inverse limits with mappings also do not carry over as homeomorphisms to the set-valued case. Instead, one shift is a mapping and the other is a set-valued function. A generalized conjugacy theorem rounds out this chapter.
W. T. Ingram

Chapter 5. Dimension

Inverse limits on [0, 1] with mappings cannot raise dimension. By using set-valued functions, however, such an inverse limit can be infinite dimensional. In this chapter, we examine aspects of dimension in inverse limits on [0, 1] with set-valued functions. We give an example of an inverse limit on [0, 1] with set-valued functions that has dimension 2 and another having dimension 3. We conclude this chapter with a proof that an inverse limit on [0, 1] with upper semicontinuous functions cannot be a 2-cell.
W. T. Ingram

Chapter 6. Problems

This chapter contains statements of some unsolved problems in the theory of inverse limits with set-valued functions. The chapter ends with a references (current at the time of publication of this book) listing all of the books and papers on this subject that are known to the author.
W. T. Ingram


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