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Spectral zeta functions of fractals and the complex dynamics of polynomials
Author(s):
Alexander
Teplyaev
Journal:
Trans. Amer. Math. Soc.
359
(2007),
4339-4358.
MSC (2000):
Primary 28A80, 37F10;
Secondary 20H05, 35P20, 37A30, 47A10, 58C40
Posted:
March 20, 2007
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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:
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
Posted:
March 20, 2007
Additional Notes:
This research was supported in part by NSF grants DMS-0071575 and DMS-0505622
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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