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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Spectral zeta functions of fractals and the complex dynamics of polynomials


Author: Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 359 (2007), 4339-4358
MSC (2000): Primary 28A80, 37F10; Secondary 20H05, 35P20, 37A30, 47A10, 58C40
Published electronically: March 20, 2007
MathSciNet review: 2309188
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Abstract: We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half-plane with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta function of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings.


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Additional Information

Alexander Teplyaev
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: teplyaev@math.uconn.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04150-5
PII: S 0002-9947(07)04150-5
Keywords: Spectral zeta function, fractal, rational complex dynamics, Laplacian, fractal string
Received by editor(s): May 27, 2005
Received by editor(s) in revised form: August 16, 2005
Published electronically: March 20, 2007
Additional Notes: This research was supported in part by NSF grants DMS-0071575 and DMS-0505622
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.