Spectral zeta functions of fractals and the complex dynamics of polynomials
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Abstract:
We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpiński 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 Sierpiński 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.References
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Additional Information
- Alexander Teplyaev
- Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
- MR Author ID: 361814
- Email: teplyaev@math.uconn.edu
- 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
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 4339-4358
- MSC (2000): Primary 28A80, 37F10; Secondary 20H05, 35P20, 37A30, 47A10, 58C40
- DOI: https://doi.org/10.1090/S0002-9947-07-04150-5
- MathSciNet review: 2309188