Analysis on Fractals

Abstract

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.

Existence of self-similar energies on finitely ramified fractals

“…i) The map (1/R) is continuous by Lemma 2.4. Moreover, it is one-to-one on Q by [4], Theorem 2.1.12. We are going to prove it is one-to-one on Q, by reducing this case to that on Q, using Lemma 2.7.…”

“…In the latter case, by Lemma 5.5 i) Π n 2 ,n 1 (ϖ) = j (n 1 ) , thus the unique possibility is that ϖ ∈ C (2) j,l for some l with j ∈ B ′ l by the definition of the C (5) . It follows that ϖ, ϖ ′ ∈ C (4) l . As, by Lemma 6.2, ϖ / ∈ C (4) l ′ if l ̸ = l ′ , then i) and ii) easily follow.…”

“…self-similar sets see [4]. Conversely, there exists a standard way to associate to a given fractal triple a self-similar fractal K, and also an n-fractal triple corresponding to the previously defined set Ψ n of one-to-one maps and generating the same fractal K. Namely, given a fractal triple F := (V (0) , V (1) , Ψ), Ψ = {ψ 1 , ..., ψ k }, we can define a related fractal, F in the following way: Let…”

“…We now discuss the structure of such sets. In particular, Lemma 6.6 i) states that C (5) can be represented as C (4) l with the addition of the separate tails C (7) j,l .…”

“…The general theory of P.C.F. self-similar sets and many examples can be found in [4]. In this paper, I essentially consider P.C.F.…”