Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

by Pertti Mattila

Overview

The focus of this book is geometric properties of general sets and measures in Euclidean spaces. Applications of this theory include fractal-type objects, such as strange attractors for dynamical systems, and those fractals used as models in the sciences. The author provides a firm and unified foundation for the subject and develops all the main tools used in its study, such as covering theorems, Hausdorff measures and their relations to Riesz capacities and Fourier transforms. The last third of the book is devoted to the Besicovitch-Federer theory of rectifiable sets, which form in a sense the largest class of subsets of Euclidean space possessing many of the properties of smooth surfaces.

Product Details

ISBN-13:	9780521655958
Publisher:	Cambridge University Press
Publication date:	02/25/1999
Series:	Cambridge Studies in Advanced Mathematics , #44
Edition description:	New Edition
Pages:	356
Product dimensions:	5.98(w) x 8.90(h) x 0.91(d)

Table of Contents

Acknowledgements; Basic notation; Introduction; 1. General measure theory; 2. Covering and differentiation; 3. Invariant measures; 4. Hausdorff measures and dimension; 5. Other measures and dimensions; 6. Density theorems for Hausdorff and packing measures; 7. Lipschitz maps; 8. Energies, capacities and subsets of finite measure; 9. Orthogonal projections; 10. Intersections with planes; 11. Local structure of s-dimensional sets and measures; 12. The Fourier transform and its applications; 13. Intersections of general sets; 14. Tangent measures and densities; 15. Rectifiable sets and approximate tangent planes; 16. Rectifiability, weak linear approximation and tangent measures; 17. Rectifiability and densities; 18. Rectifiability and orthogonal projections; 19. Rectifiability and othogonal projections; 19. Rectifiability and analytic capacity in the complex plane; 20. Rectifiability and singular intervals; References; List of notation; Index of terminology.

Editorial Reviews

"Provides a unified theory for the study of the topic and develops the main tools used in its study including theorems, Hausdorff measures, and their relations to Riesz capacities and Fourier transforms." Book News, Inc.

From the Publisher

Provides a unified foundation for the study of the topic and develops the main tools used, including theorems, Hausdorff measures, and their relations to Riesz capacities and Fourier transforms. The last section of the book is devoted to the Besicovitch-Federed theory of rectifiable sets. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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