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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 $\rho \left(\lambda \right)$ behaves so wildly that $\rho \left(\lambda \right)$ does not vary regularly, and the ratio $\rho \left(\lambda \right)/{\lambda }^{{d}_{s}/2}$ is bounded but not convergent as $\lambda ↗\infty$, where ${d}_{s}$ is the spectral dimension of the p.c.f. self-similar set.

##### MSC:
 58J50 Spectral problems; spectral geometry; scattering theory 37A30 Ergodic theorems, spectral theory, Markov operators
##### References:
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