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Zbl 0861.58047
On eigenvalue problems for Laplacians on p.c.f. self-similar sets.
(English)
[J] Japan J. Ind. Appl. Math. 13, No.1, 1-23 (1996). ISSN 0916-7005; ISSN 1868-937X/e

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 $\rho(\lambda)$ behaves so wildly that $\rho(\lambda)$ does not vary regularly, and the ratio $\rho(\lambda)/\lambda^{d_s/2}$ is bounded but not convergent as $\lambda \nearrow \infty$, where $d_s$ is the spectral dimension of the p.c.f. self-similar set.
MSC 2000:
*58J50 Spectral problems; spectral geometry; scattering theory
37A30 Ergodic theorems, spectral theory, Markov operators

Keywords: eigenvalue; post critically finite self-similar sets; spectral decimation; spectral dimension; Laplacian

Cited in: Zbl 1177.28029

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