Harmonic Calculus on P.C.F. Self-Similar Sets

On singularity of energy measures on self-similar sets

“…For i ∈ S, define a shift operator σ i : → by σ i (ω 1 ω 2 · · · ) = iω 1 ω 2 · · · . Let us suppose that there exists a continuous surjective map π : → K such that ψ i • π = π • σ i for each i ∈ S. We call (K, S, {ψ i } i∈S ) a self-similar structure, following Kigami [15].…”

Criteria for Spectral Gaps of Laplacians on Fractals

“…fractals [11,12] and the spectral decimation method to analyze their spectra developed by Shima [16].…”

“…The (normalized) Laplacian on K can be defined as a limit of the normalized discrete Laplacians m [11] and [12].…”

“…By Theorem 4.10 in [11], it is known that for some s ∈ S, r s < K D (0) −1 . Therefore K D (0) < 1 and |S| < 1 K D (0)r 0 .…”

A generalization of Hata-Yamaguti’s results on the Takagi function II: Multinomial case

“…We only remark that if r --1/2, that is, #~ is the Lebesgue measure, then Sn(x) = r We also note that the above system (3), in a sense, corresponds to Poisson equations with boundary conditions and plays an important role in harmonic analysis on fractal sets. See Kigami [6], [7], and Yamaguti, Hata and Kigami…”