Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities. (English) Zbl 1396.28017
Summary: We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps are in certain sense essentially of finite type, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain a closed formula for the spectral dimension of the Laplacian defined by a self-similar measure satisfying this condition. For Laplacians defined by fractal measures with overlaps, spectral dimension has been obtained earlier only for a small class of one-dimensional self-similar measures satisfying Strichartz second-order self-similar identities. The main technique we use relies on the vector-valued renewal theorem proved by K.-S. Lau et al. [Stud. Math. 117, No. 1, 1–28 (1995; Zbl 0839.43002)].
MSC:
| 28A80 | Fractals |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 43A05 | Measures on groups and semigroups, etc. |
| 47A75 | Eigenvalue problems for linear operators |
Keywords:
fractal; Laplacian; spectral dimension; self-similar measure with overlaps; essentially of finite typeCitations:
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