Hyperfunctions and spectral zeta functions of Laplacians on self-similar fractals

Abstract

We investigate the spectral zeta function of fractal differential operators such as the Laplacian on the unbounded (i.e. infinite) Sierpinski gasket and a self-similar Sturm–Liouville operator associated with a fractal self-similar measure on the half-line. In the latter case, Sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables, called the renormalization map. We obtain a factorization of the spectral zeta function of such an operator, expressed in terms of the Dirac delta hyperfunction, a geometric zeta function, and the zeta function associated with the dynamics of the corresponding renormalization map, viewed either as a polynomial function on (in the first case) or (in the second case) as a polynomial on the complex projective plane, . Our first main result extends to the case of the fractal Laplacian on the unbounded Sierpinski gasket a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended by Teplyaev to the bounded (i.e. finite) Sierpinski gasket and some other decimable fractals. Furthermore, our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal Sturm–Liouville operators. Moreover, as a corollary, in the very special case when the underlying self-similar measure is the Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in , thereby extending to several variables an analogous result by Teplyaev.