This book covers analysis on fractals, a developing area of mathematics that 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.
Inhaltsverzeichnis
Introduction; 1. Geometry of self-similar sets; 2. Analysis on limits of networks; 3. Construction of Laplacians on P. C. F. self-similar structures; 4. Eigenvalues and eigenfunctions of Laplacians; 5. Heat kernels; Appendix A: Additional fact; Appendix B: Mathematical backgrounds; Bibliography; List of notations; Index.
