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.

We establish an analogue of WeyΓs classical theorem for the asymptotics of eigenvalues of Laplacians on a finitely ramified (i.e., p.c.f.) self-similar fractal K, such as, for example, the Sierpinski gasket. We consider both Dirichlet and Neumann boundary conditions, as well as Laplacians associated with Bernoulli-type ("multifractal") measures on K. From a physical point of view, we study the density of states for diffusions or for wave propagation in fractal media. More precisely, let Q(X) be the number of eigenvalues less than x. Then we show that ρ(x) is of the order of x ds / 2 as x -» +00, where the "spectral exponent" d s is computed in terms of the geometric as well as analytic structures of K. Further, we give an effective condition that guarantees the existence of the limit of x~d s / 2 ρ(x) as x -> -hoc; this condition is, in some sense, "generic". In addition, we define in terms of the above "spectral exponents" and calculate explicitly the "spectral dimension" of K.

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A reconstruction of the well-known theory of calculus on [0, 1] will naturally bring a calculus associated with the harmonic functions, Laplace operator, Gauss-Green's formula and so on, on the N-Sierpinski space whose Hausdorff dimension is (log N)/(log 2).

In this paper, we define the Green functions for a resistance form by using effective resistance and harmonic functions. Then the Green functions and harmonic functions are shown to be uniformly Lipschitz continuous with respect to the resistance metric. Making use of this fact, we construct the Green operator and the (measure valued) Laplacian. The domain of the Laplacian is shown to be a subset of uniformly Lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1/2-Hölder continuous functions.