Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

Michael Levitin; Dmitri Vassiliev

doi:10.1112/plms/s3-72.1.188

Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

Michael Levitin
, Dmitri Vassiliev

Research output: Contribution to journal › Article › peer-review

61 Citations (Scopus)

Abstract

We consider the asymptotic behaviour of the volume of the Minkowski sausage, the counting function of the Dirichlet Laplacian, the partition function and the heat content for an iterated set script G sign ? R^m with fractal boundary. We show, using the renewal theory (well known in probability) that in all cases the asymptotic behaviour depends essentially on whether the set of logarithms of the similitudes used in the construction of script G sign is arithmetic.

Original language	English
Pages (from-to)	188-214
Number of pages	27
Journal	Proceedings of the London Mathematical Society
Volume	72
Issue number	1
DOIs	https://doi.org/10.1112/plms/s3-72.1.188
Publication status	Published - Jan 1996

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abstract = "We consider the asymptotic behaviour of the volume of the Minkowski sausage, the counting function of the Dirichlet Laplacian, the partition function and the heat content for an iterated set script G sign ? Rm with fractal boundary. We show, using the renewal theory (well known in probability) that in all cases the asymptotic behaviour depends essentially on whether the set of logarithms of the similitudes used in the construction of script G sign is arithmetic.",

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AB - We consider the asymptotic behaviour of the volume of the Minkowski sausage, the counting function of the Dirichlet Laplacian, the partition function and the heat content for an iterated set script G sign ? Rm with fractal boundary. We show, using the renewal theory (well known in probability) that in all cases the asymptotic behaviour depends essentially on whether the set of logarithms of the similitudes used in the construction of script G sign is arithmetic.

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JO - Proceedings of the London Mathematical Society

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Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals

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