Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem
We consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Ou...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
2003
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| Summary: | We consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of the Laplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling in the eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to do this we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for the eigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth. |
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