Spectral zeta functions of fractals and the complex dynamics of polynomials

• Autores: Alexander Teplyaev
• Localización: Transactions of the American Mathematical Society, ISSN 0002-9947, Vol. 359, Nº 9, 2007, págs. 4339-4358
• Idioma: inglés
• DOI: 10.1090/s0002-9947-07-04150-5
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• Resumen

  • 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.