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Nevertheless, each point is a branching point in the sense that it can be connected by three disjoint curves to the three vertices of the triangle. The Sierpinski gasket in gure 1 has topological dimension one. A simple curve can be nowhere dierentiable, it can even have positive area. A Cantor set is uncountable yet contains no connected pieces. They were designed to clarify basic concepts like 'continuum', 'curve' and 'one-dimensional space', by demonstrating that subsets of the line or plane can have very strange, counterintuitive properties. 1 Introduction Most of the well-known examples of fractals come from the early stage of set theory and general topology at about 1900. We construct some new types of Sierpinski gasket and determine their exponent of interior distance. After an introduction to self-similar deterministic and random measures, we introduce an algorithm which decides on the separation condition for given similarity mappings dening a self-similar set. Several books on this topic are available which have become popular even amo. Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper, gained more and more interest. Keywords: Fractal geometry, iterated function systems, complete metric spaces, Baire space, Hausdorff measure, Hausdorff dimension, selfsimilarity. On the other hand, there are closed and bounded non-empty sets not describable by IIFS. Comparing the descriptive power of IFS and IIFS as mechanisms defining closed and bounded sets, we show that IIFS are strictly more powerful than IFS.
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Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. We state results analogous to the well-known case of finitely many mappings (IFS). These ideas have considerable scope for further development, and a list of problems and lines of research is included.: We examine iterated function systems consisting of a countably infinite number of contracting mappings (IIFS). Halmos, 1950, Graduate Texts in Mathematics, 18, Springer New York. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a. 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. 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 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. 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. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before.
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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. 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.