Fractals and Universal Spaces in Dimension Theory

Hardcover | December 3, 2008

byStephen Lipscomb

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Historically, 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 metric).The classical theory is now well documented in several books. This monograph is the first book to unify the modern theory from 1960-2007. Like the classical theory, the modern theory fundamentally involves the unit interval.Unique features include:* The use of graphics to illustrate the fractal view of these spaces;* Lucid coverage of a range of topics including point-set topology and mapping theory, fractal geometry, and algebraic topology;* A final chapter contains surveys and provides historical context for related research that includes other imbedding theorems, graph theory, and closed imbeddings;* Each chapter contains a comment section that provides historical context with references that serve as a bridge to the literature.This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. Being the first monograph to focus on the connection between generalized fractals and universal spaces in dimension theory, it will be a natural text for graduate seminars or self-study - the interested reader will find many relevant open problems which will create further research into these topics.

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From the Publisher

Historically, 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 metric).The classical theory is now well documented in several books. Thi...

From the Jacket

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 metric). While the classical theory is now well documented in several books, this is th...

Format:HardcoverDimensions:260 pages, 9.25 × 6.1 × 0.04 inPublished:December 3, 2008Publisher:Springer New YorkLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0387854932

ISBN - 13:9780387854939

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Table of Contents

Preface.- Introduction.- Construction of JA=j alpha.- Self-Similarity and Jn+1 for Finite n.- No-Carry Property of wA.- Imbedding Ja in Hilbert Space.- Infinite IFS with Attractor wA.- Infinite IFS with Attractor wA.- Dimension Zero.- Decompositions.- The Jn+1 Imbedding Theorem.- Minimal-Exponent Question.- The JA Imbedding Theorem.- 1992-2007 Ja Related Research.- Isotopy Moves J5 Into 3-Space.- From 2-Web IFS to 2 Simplex IFS 2-Space and the 1-Sphere.- From 3-Web IFS to 3-Simplex 3-Space and the 2-Sphere.- Background Basics.- The Standard Simplex.- Measures and Fractal Dimension.- Bibliography.- Index.

Editorial Reviews

From the reviews:"The book is a research monograph reporting on an interesting area of research arising from the confluence of two streams: topology and self-similar fractals. It is written at a level that could be understood by graduate students and advanced undergraduates. It could be used for a seminar or introductory course for either topology or self-similar sets. . The historical notes are informative and interesting. There is an extensive bibliography at the end of the book documenting the results and historical comments." (J. E. Keesling, Mathematical Reviews, Issue 2011 b)"The book under review is devoted to dimension theory in general. . The book is completed by a useful appendix consisting of three parts, devoted to basics in topology, standard simplices in Hilbert spaces, and fractal geometry. So, it is accessible for all mathematicians, but should be of special interest for those working in topology or fractal geometry. The book contains a remarkable number of interesting historical remarks and colorful pictures." (Uta Freiberg, Zentralblatt MATH, Vol. 1210, 2011)