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Springer Monographs in Mathematics

Fractals and Universal Spaces in Dimension Theory

Authors: Lipscomb, Stephen

  • Includes more than 60 black-and-white illustrations and 15 in color
  • Each chapter contains comments that provide historical context
  • This is the first time most of this material has appeared in book form
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eBook $89.00
price for USA (gross)
  • ISBN 978-0-387-85494-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $119.00
price for USA
  • ISBN 978-0-387-85493-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $119.00
price for USA
  • ISBN 978-1-4419-2751-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

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 the first book to unify the modern theory (1960 – 2007). Like the classical theory, the modern theory fundamentally involves the unit interval.

 

By the 1970s, the author of this monograph generalized Cantor’s 1883 construction (identify adjacent-endpoints in Cantor’s set) of the unit interval, obtaining — for any given weight — a one-dimensional metric space that contains rationals and irrationals as counterparts to those in the unit interval.

 

Following the development of fractal geometry during the 1980s, these new spaces turned out to be the first examples of attractors of infinite iterated function systems — “generalized fractals.” The use of graphics to illustrate the fractal view of these spaces is a unique feature of this monograph.  In addition, this book provides historical context for related research that includes imbedding theorems, graph theory, and closed imbeddings.

 

This monograph will be useful to topologists, to mathematicians working in fractal geometry, and to historians of mathematics. It can also serve as a text for graduate seminars or self-study — the interested reader will find many relevant open problems that will motivate further research into these topics.

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)


Table of contents (14 chapters)

  • Decompositions1

    Lipscomb, Stephen Leon

    Pages 1-24

  • Isotopy Moves J 5 into 3-Space

    Lipscomb, Stephen Leon

    Pages 1-12

  • Construction of J A = J α

    Lipscomb, Stephen Leon

    Pages 1-10

  • From 3-Web IFS to 3-Simplex IFS 3-Space and the 2-Sphere

    Lipscomb, Stephen Leon

    Pages 1-35

  • Dimension Zero

    Lipscomb, Stephen Leon

    Pages 1-6

Buy this book

eBook $89.00
price for USA (gross)
  • ISBN 978-0-387-85494-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $119.00
price for USA
  • ISBN 978-0-387-85493-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $119.00
price for USA
  • ISBN 978-1-4419-2751-4
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Fractals and Universal Spaces in Dimension Theory
Authors
Series Title
Springer Monographs in Mathematics
Copyright
2009
Publisher
Springer-Verlag New York
Copyright Holder
Springer-Verlag New York
eBook ISBN
978-0-387-85494-6
DOI
10.1007/978-0-387-85494-6
Hardcover ISBN
978-0-387-85493-9
Softcover ISBN
978-1-4419-2751-4
Series ISSN
1439-7382
Edition Number
1
Number of Pages
XVIII, 242
Number of Illustrations and Tables
76 b/w illustrations, 15 illustrations in colour
Topics