Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals

TL;DR

This paper explores the spectral zeta functions of Laplacians on fractals like the Sierpinski gasket, revealing new factorizations involving hyperfunctions and complex dynamics, and connects these to the Riemann zeta function.

Contribution

It extends spectral zeta function factorizations to unbounded fractals and generalizes to complex renormalization maps, linking fractal spectra with complex dynamics and hyperfunctions.

Findings

Factorization of spectral zeta functions using hyperfunctions and dynamical zeta functions

Extension of known formulas to unbounded fractals like the Sierpinski gasket

Representation of the Riemann zeta function via polynomial dynamics in complex projective space

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, C. 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 the complex plane (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 A. 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 Lebesgue measure on [0, 1], we obtain a representation of the Riemann zeta function in terms of the dynamics of a certain polynomial in the complex projective plane, thereby extending to several variables an analogous result by A. Teplyaev.