Analysis on Fractals

By: Jun Kigami, Bela Bollobas (Editor), W. Fulton (Editor)

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Hardcover | 12 December 2001

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Hardcover
236 Pages

Dimensions(cm)
22.86 x 15.24 x 1.75

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This book covers analysis on fractals, a developing area of mathematics which focuses on the dynamical aspects of fractals, such as heat diffusion on fractals and the vibration of a material with fractal structure. The book provides a self-contained introduction to the subject, starting from the basic geometry of self-similar sets and going on to discuss recent results, including the properties of eigenvalues and eigenfunctions of the Laplacians, and the asymptotical behaviors of heat kernels on self-similar sets. Requiring only a basic knowledge of advanced analysis, general topology and measure theory, this book will be of value to graduate students and researchers in analysis and probability theory. It will also be useful as a supplementary text for graduate courses covering fractals.

Industry Reviews

'... the most recent introduction to the analysis of 'Laplacians' on what physicists call finitely ramified self-similar fractals.' Volker Metz, Zentralblatt MATH 'Anyone with a background in the analysis of linear field equations, with an interest in heterogeneous media, or who is looking to breathe new life into their research, should read this book.' A. J. Mulholland, Proceedings of the Edinburgh Mathematical Society 'This book is an introduction to the subject written by one of the active researchers in the area. It is recommended to those who would like to go from the basics to current research topics.' Acta. Sci. Math.

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Introduction	p. 1
Geometry of Self-Similar Sets	p. 8
Construction of self-similar sets	p. 8
Shift space and self-similar sets	p. 12
Self-similar structure	p. 17
Self-similar measure	p. 25
Dimension of self-similar sets	p. 28
Connectivity of self-similar sets	p. 33
Analysis on Limits of Networks	p. 41
Dirichlet forms and Laplacians on a finite set	p. 41
Sequence of discrete Laplacians	p. 51
Resistance from and resistance metric	p. 55
Dirichlet forms and Laplacians on limits of networks	p. 63
Construction of Laplacians on P. C. F. Self-Similar Structures	p. 68
Harmonic structures	p. 69
Harmonic functions	p. 73
Topology given by effective resistance	p. 83
Dirichlet forms on p. c. f. self-similar sets	p. 88
Green's function	p. 94
Green's operator	p. 102
Laplacians	p. 107
Nested fractals	p. 115
Eigenvalues and Eigenfunctions of Laplacians	p. 131
Eigenvalues and eigenfunctions	p. 132
Relation between dimensions	p. 137
Localized eigenfunctions	p. 141
Existence of localized eigenfunctions	p. 146
Estimate of eigenfunctions	p. 152
Heat Kernels	p. 157
Construction of heat kernels	p. 158
Parabolic maximum principle	p. 164
Asymptotic behavior of the heat kernels	p. 171
Additional Facts	p. 180
Mathematical Background	p. 196
Bibliography	p. 212
Index of Notation	p. 221
Index	p. 222
Table of Contents provided by Blackwell. All Rights Reserved.

Analysis on Fractals

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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Other Editions and Formats

Paperback

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