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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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