Fractals and Universal Spaces in Dimension Theory
For metric spaces the quest for universal spaces in dimension theory spanned approximately a century of mathematical research. The history breaks naturally into two periods — the classical (separable metric) and the modern (not necessarily separable metri
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Stephen Leon Lipscomb
Fractals and Universal Spaces in Dimension Theory With 91 Illustrations and 10 Tables
123
Stephen Leon Lipscomb Emeritus Professor of Mathematics Department of Mathematics University of Mary Washington Fredericksburg, VA 22401 USA [email protected]
ISSN: 1439-7382 ISBN: 978-0-387-85493-9 DOI 10.1007/978-0-387-85494-6
e-ISBN: 978-0-387-85494-6
Library of Congress Control Number: 2008938816 Mathematics Subject Classification (2000): 54xx, 28A80, 57xx c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper springer.com
Dedicated to my wife Patty, our sons Stephen and Darrin, and my mother Dema Ann (Alkire), and the memory of my father David Leon Lipscomb
Balsa-wood model of J5 constructed by Gene Miller.1 1 Photograph
by Marlin Thomas; graphical adjustments by Bulent Atalay.
Contents Preface Introduction
xi xiii
Chapter 1. Construction of JA = Jα §1 Baire’s Zero-Dimensional Spaces §2 Adjacent-Endpoint Relation §3 JA and the Natural Map p §4 Comments
1 1 4 5 6
Chapter 2. Self-Similarity and Jn+1 for Finite n §5 Self-Similarity of JA §6 Approximations for n + 1 = 2, 3, 4 §7 Approximations for n + 1 = 5 §8 Jn+1 as an Attractor ω n of an IFS §9 Can We “View” Jn+1 in 3-Space? §10 Comments
11 11 12 13 16 20 21
Chapter 3. No-Carry Property of ω A §11 Three Examples §12 Star Spaces §13 The Star Space in l2 (A) §14 Projecting N (A) onto a Cantor-Star Subspace §15 Projecting JA onto a Star Subspace §16 Mapping JA into l2 (A ) §17 No-Carry Characterization of ω A §18 Comments
23 23 25 26 27 27 28 29 31
Chapter 4. Imbedding JA in Hilbert Space §19 Characterization of the Adjacent-Endpoint Relation §20 The Mapping f : JA → ω A §21 Sierpi´ nski’s Recursive Construction §22 Milutinovi´c’s Subspace MA of Hilbert Space §23 Comments
33 33 35 38 39 40
Chapter 5. Infinite IFS with Attractor ω A §24 Neighborhoods of Sets §25 Hausdorff Metrics and Pseudo Metrics §26 Completeness of (BX , h) §27 Hutchinson Operator for a Bounded IFS vii
41 41 42 44 47
viii
CONTENTS
§28 §29 §30 §31
The Attractor of an Infinite IFS The JA System The JA System Has Attractor ω A Comments
Chapter 6. Dimension Zero §32 §33 §34 §35
Rationals and Irrationals n+1 Imbedding Theorem for n = 0 JA Subspaces of JA Comments
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