The Geometry of Fractal Sets (Cambridge Tracts in Mathematics, Series Number 85)

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.

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.

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).