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On eigenvalue problems for Laplacians on p.c.f. self-similar sets. (English) Zbl 0861.58047
This paper considers 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. The author shows that, with some additional assumptions, the eigenvalue counting function ρ(λ) behaves so wildly that ρ(λ) does not vary regularly, and the ratio ρ(λ)/λ d s /2 is bounded but not convergent as λ, where d s is the spectral dimension of the p.c.f. self-similar set.

MSC:
58J50Spectral problems; spectral geometry; scattering theory
37A30Ergodic theorems, spectral theory, Markov operators
References:
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