Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the s...
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| Format: | Journal article |
| Language: | English |
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2006
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| _version_ | 1797083951372697600 |
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| author | Hambly, B Lapidus, M |
| author_facet | Hambly, B Lapidus, M |
| author_sort | Hambly, B |
| collection | OXFORD |
| description | In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary. ©2005 American Mathematical Society. |
| first_indexed | 2024-03-07T01:48:48Z |
| format | Journal article |
| id | oxford-uuid:995a0c9a-3eba-4c4c-8e10-56198f8b58e3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T01:48:48Z |
| publishDate | 2006 |
| record_format | dspace |
| spelling | oxford-uuid:995a0c9a-3eba-4c4c-8e10-56198f8b58e32022-03-27T00:13:45ZRandom fractal strings: Their zeta functions, complex dimensions and spectral asymptoticsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:995a0c9a-3eba-4c4c-8e10-56198f8b58e3EnglishSymplectic Elements at Oxford2006Hambly, BLapidus, MIn this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary. ©2005 American Mathematical Society. |
| spellingShingle | Hambly, B Lapidus, M Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title | Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title_full | Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title_fullStr | Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title_full_unstemmed | Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title_short | Random fractal strings: Their zeta functions, complex dimensions and spectral asymptotics |
| title_sort | random fractal strings their zeta functions complex dimensions and spectral asymptotics |
| work_keys_str_mv | AT hamblyb randomfractalstringstheirzetafunctionscomplexdimensionsandspectralasymptotics AT lapidusm randomfractalstringstheirzetafunctionscomplexdimensionsandspectralasymptotics |