Spectral asymptotics for Laplacians on self-similar sets |
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Authors: | Naotaka Kajino |
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Affiliation: | Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan |
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Abstract: | Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function. |
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Keywords: | Self-similar sets Dirichlet forms Eigenvalue counting function Partition function Short time asymptotics Sub-Gaussian heat kernel estimate Sierpinski carpets |
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