Stephen Lipscomb

Fractals and Universal Spaces in Dimension Theory (eBook, PDF)

• Format: PDF

Produktbeschreibung

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.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 242
• Erscheinungstermin: 28. Oktober 2008
• Englisch
• ISBN-13: 9780387854946
• Artikelnr.: 37288803

Inhaltsangabe

Construction of =.- Self-Similarity and for Finite.- No-Carry Property of.- Imbedding in Hilbert Space.- Infinite IFS with Attractor.- Dimension Zero.- Decompositions.- The Imbedding Theorem.- Minimal-Exponent Question.- The Imbedding Theorem.- 1992#x2013;2007 -Related Research.- Isotopy Moves into 3-Space.- From 2-Web IFS to 2-Simplex IFS 2-Space and the 1-Sphere.- From 3-Web IFS to 3-Simplex IFS 3-Space and the 2-Sphere.

Rezensionen

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)