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 cert
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ziirich F. Takens, Groningen
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Keith R. Wicks
Fractals and Hyperspaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Keith R. Wicks Department of Mathematics and Computer Science University College of Swansea Singleton Park Swansea SA2 8PP, U. K.
The picture on the front cover shows a zoom-in on Fig. 14, page 51
Mathematics Subject Classification (1991): 03H05, 05B45, 51N05, 52A20, 52A45, 54A05, 54B20, 54C60, 54E35, 54E40, 54H05, 54H20, 54H25, 54J05
ISBN 3-540-54965-X Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54965-X Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 199] Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 - Printed on acid-free paper
Foreword The main theme of this monograph is the study of fractals and fractal notions, backed up by a self-contained nonstandard development of relevant hyperspace theory, particularly as regards the Hausdorff metric and Vietoris topology. The fractal study itself is in two parts, the first developing and making contributions to the theory of J. E. Hutchinson's invariant sets, sets which are self-similar in the sense of being composed of smaller images of themselves. The second part explores newer territory, introducing the formal notion of a 'view' as part of a general framework concerned with studying the structure and perception of sets within a given space, and in particular we use views to express and investigate new concepts of self-similarity and fractality which are then considered in connection with invariant sets, a large class of which are shown to be 'visually fractal' in a certain precise sense. Complete with many figures and suggestions for further work, the monograph should be of relevance to those interested in fractals, hyperspaces, fixed-point theory, tilings, or nonstandard analysis. The work was undertaken at the University of Hull during the period 1987-90, financed for two years by an SERC research grant which I gratefully acknowledge. My grateful thanks go also to Professor Nigel Outland for assistance and advice throughout, and to Dr. Dona Strauss for help with many topological and other queries. Keith R. Wicks Hull, August 1991.
Contents Page
Introduction
1
Preliminaries
3
Chapter 1: Nonstandard De
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