The zeta function of the Laplacian on certain fractals

Gregory Derfel; Peter J. Grabner; Fritz Vogl

doi:10.1090/S0002-9947-07-04240-7

The zeta function of the Laplacian on certain fractals

Gregory Derfel, Peter J. Grabner, Fritz Vogl

Department of Mathematics

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

23 Scopus citations

Abstract

We prove that the zeta function ζΔ of the Laplacian Δ on selfsimilar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function, thereby answering a question posed by J. Kigami and M. Lapidus for this class of fractals.

Original language	English
Pages (from-to)	881-897
Number of pages	17
Journal	Transactions of the American Mathematical Society
Volume	360
Issue number	2
DOIs	https://doi.org/10.1090/S0002-9947-07-04240-7
State	Published - 1 Feb 2008

Keywords

Complex dynamics
Dirichlet series
Fractals
Laplace operator
Spectral decimation

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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10.1090/S0002-9947-07-04240-7

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KW - Laplace operator

KW - Spectral decimation

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The zeta function of the Laplacian on certain fractals

Abstract

Keywords

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