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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