Abstract
We formulate and study a strong harmonic structure under which eigenvalues of the Laplacian on a p.c.f. self-similar set are completely determined according to the dynamical system generated by a rational function. We then show that, with some additional assumptions, the eigenvalue counting function ρ(λ) behaves so wildly that ρ(λ) does not vary regularly, and the ratio\(\rho (\lambda )/\lambda ^{d_s /2} \) is bounded but non-convergent as λϖ∞, whered s is the spectral dimension of the p.c.f. self-similar set.
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Shima, T. On eigenvalue problems for Laplacians on P.C.F. self-similar sets. Japan J. Indust. Appl. Math. 13, 1–23 (1996). https://doi.org/10.1007/BF03167295
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DOI: https://doi.org/10.1007/BF03167295