The Geometry of Fractal Sets

K. J. Falconer (University of Bristol), B. Bollobas, W. Fulton , A. Katok

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English

Cambridge University Press
22 September 1986

Calculus & mathematical analysis; Analytic topology

Series: Cambridge Tracts in Mathematics

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.

By: K. J. Falconer (University of Bristol)
Series edited by: B. Bollobas, W. Fulton, A. Katok, F. Kirwan
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Volume: 85
Dimensions: Height: 228mm, Width: 154mm, Spine: 11mm
Weight: 270g
ISBN: 9780521337052
ISBN 10: 0521337054
Series: Cambridge Tracts in Mathematics
Pages: 180
Publication Date: 22 September 1986
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Preface; Introduction; Notation; 1. Measure and dimension; 2. Basic density properties; 3. Structure of sets of integral dimension; 4. Structure of sets of non-integral dimension; 5. Comparable net measures; 6. Projection properties; 7. Besicovitch and Kakeya sets; 8. Miscellaneous examples of fractal sets; References; Index.

Reviews for The Geometry of Fractal Sets

'This book is filled with the geometric jewels of fractional and integral Hausdorff dimension. It contains a much-needed unified notation and includes many recent results with simplified proofs. The theory is classically and rigorously presented with applications only alluded to in the introduction. Each chapter contains a short and important problem set. This is a lovely introduction to the mathematics of fractal sets for the pure mathematician.' American Mathematical Monthly ...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