Fractals and Hyperspaces
Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The theory of J.E. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. This leads to new, elegant concepts (defined purely topologically) of self-similarity and fractality: in particular, the author shows that many invariant sets are "visually fractal", i.e. have infinite detail in a certain sense. These ideas have considerable scope for further development, and a list of problems and lines of research is included.
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Section Page 0 Introduction
Monads of Subsets
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attractor bijective body topology bounded boundedly compact Chapter closed sets closed subsets compact Hausdorff space compact iff compact set condition compact-open topology compact-uniform topology continuous map contraction mapping theorem convex Corollary defined dense diam direct similitudes disjoint elements of F embedded equivalently example F satisfies fixed point fractal gives Hausdorff metric Hausdorff space hence homeomorphism homeomorphism condition infinite infinitesimal intersects invariant sets isometries limit view locally compact metric space monads monoid nearstandard nonempty compact set nonempty view nonoverlapping nonstandard analysis norm Note nth-level images product topology Proof Proposition regular residual respect to F result S-compact topology satisfies the compact Section self-similar semiflow Semigroup Semigroup hF set F set of contractions similarity view structure subspace there's tiled with respect topological monoid topological space Vietoris topology view class view domain view topology