The Geometry of Fractal Sets (Cambridge Tracts in Mathematics) [Paperback]

K. J. Falconer (Author)

Book Description

Publication Date: July 25, 1986 | ISBN-10: 0521337054 | ISBN-13: 978-0521337052

This book contains a rigorous mathematical treatment of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Questions of local density and the existence of tangents of such sets are studied, as well as the dimensional properties of their projections in various directions. In the case of sets of integral dimension the dramatic differences between regular 'curve-like' sets and irregular 'dust like' sets are exhibited. The theory is related by duality to Kayeka sets (sets of zero area containing lines in every direction). The final chapter includes diverse examples of sets to which the general theory is applicable: discussions of curves of fractional dimension, self-similar sets, strange attractors, and examples from number theory, convexity and so on. There is an emphasis on the basic tools of the subject such as the Vitali covering lemma, net measures and Fourier transform methods.

Editorial Reviews

Review

"...by far the most accessible mathematical account available and therefore is an invaluable addition to the literature." Science

"...a lovely introduction to the mathematics of fractal sets for the pure mathematician." American Mathematical Monthly

Book Description

A mathematical study of the geometrical aspects of sets of both integral and fractional Hausdorff dimension. Considers questions of local density, the existence of tangents of such sets as well as the dimensional properties of their projections in various directions.

Product Details

• Paperback: 180 pages
• Publisher: Cambridge University Press (July 25, 1986)
• Language: English
• ISBN-10: 0521337054
• ISBN-13: 978-0521337052
• Product Dimensions: 9 x 5.8 x 0.6 inches
• Shipping Weight: 10.6 ounces (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #731,369 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

15 of 15 people found the following review helpful:

5.0 out of 5 stars Advanced treatise on fractal geometry., April 13, 2000

By 

Bernardo Vargas - See all my reviews

This review is from: The Geometry of Fractal Sets (Cambridge Tracts in Mathematics) (Paperback)

This text is a must-reading for anyone seeking advanced knowledge on fractal geometry. It is dense and deep, but clear and concise. It includes a lot of interesting material ranging from basic measure-theoretic concepts up to the disprove of Vitushkin's conjecture. It's got an extensive list of references, mostly to the original papers, making it a fundamental research tool.

As it can be inferred from the preceeding paragraph, the book is not for begineers; it was designed for graduate level courses. Undergrads and laymen should start with Edgar's "Measure, Topology, and Fractal Geometry" and Falconer's "Fractal Geometry: Mathematical Foundations and Applications".

Please check my other reviews (just click on my name above).

5 of 5 people found the following review helpful:

5.0 out of 5 stars Ingenious Compilation of Essential Fractals, September 26, 2003

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[e-mail] "[Cellular Neural N... (The Cellular Matrix) - See all my reviews

This review is from: The Geometry of Fractal Sets (Cambridge Tracts in Mathematics) (Paperback)

The Geometry of Fractal Sets by Falconer is an elegant composition of many necessary fractals, measures, projections, and dimensions. Included in the monograph are the most inspiring and applicable Besicovitch fractal sets, Kakeya fractal sets, the Appolonian packing fractal, osculatory packings, horseshoe fractals, Perron trees, hypercycloids, the Nikodym set, Lebesgue measure, Hausdorff dimension, sets of integral and non-integral dimension, sets in higher-dimensions, Borel measure, binary sets, Vitali coverings, polar reciprocity, Souslin sets, sigma-fields, tangents, net measures, the semicontinuity theorems of Golab and Vishtukin, osculatory packings, diophantine approximations, Fourier series, transforms and multipliers, Brownian motion, Grassmanian manifolds.......you name it this book explains and connects it all.

The text is written in full proper-fonting and contains many illustrations. Qualitatively the book should be of high value to researchers, graduates, and Phd's with the finest tastes.

3 of 3 people found the following review helpful:

5.0 out of 5 stars Introduction to geometric measure theory, August 22, 2003

By 

Mathieu Dutour (Jerusalem) - See all my reviews

This review is from: The Geometry of Fractal Sets (Cambridge Tracts in Mathematics) (Paperback)

This book is devoted to the hausdorf measure and Hausdorff dimension of subsets of R^n and to an extensive study of their geometry: existence of tangency, projection, etc. One chapter deals with Besicovich sets used for constructing counter-examples, especially in Harmonic analysis.

The book finish with a magnificent list of examples of haussdorff dimension computation: self-similar sets, Apollonian packings, number theory, Feigenbaum logistic map and Brownian motion.
The bibliography, of incredible quality, achieves to make the book a reference for anyone interested in fractals.