On eigenvalue problems for Laplacians on p.c.f. self-similar sets
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.
post critically finite self-similar sets
scientific article; zbMATH DE number 927306