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...
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| Format: | Journal article |
| Language: | English |
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2003
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| _version_ | 1826306496008290304 |
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| author | Hambly, B Nyberg, S |
| author_facet | Hambly, B Nyberg, S |
| author_sort | Hambly, B |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T06:48:50Z |
| format | Journal article |
| id | oxford-uuid:fbcbdfb5-3d5b-4ac2-8797-4ab8983ac2bc |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:48:50Z |
| publishDate | 2003 |
| record_format | dspace |
| spelling | oxford-uuid:fbcbdfb5-3d5b-4ac2-8797-4ab8983ac2bc2022-03-27T13:16:22ZFinitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theoremJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fbcbdfb5-3d5b-4ac2-8797-4ab8983ac2bcEnglishSymplectic Elements at Oxford2003Hambly, BNyberg, SWe 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. |
| spellingShingle | Hambly, B Nyberg, S Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title | Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title_full | Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title_fullStr | Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title_full_unstemmed | Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title_short | Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem |
| title_sort | finitely ramified graph directed fractals spectral asymptotics and the multidimensional renewal theorem |
| work_keys_str_mv | AT hamblyb finitelyramifiedgraphdirectedfractalsspectralasymptoticsandthemultidimensionalrenewaltheorem AT nybergs finitelyramifiedgraphdirectedfractalsspectralasymptoticsandthemultidimensionalrenewaltheorem |