Spectral zeta functions of fractals and the complex dynamics of polynomials

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 plain with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions 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.

References in corpus (1)

• Fractality, Self-Similarity and Complex Dimensions

Cited by in corpus (30)

• Geometry and field theory in multi-fractional spacetime
• Thermodynamics of photons on fractals
• Multifractional theories: an unconventional review
• Multi-scale gravity and cosmology
• Laplace Operators on Fractals and Related Functional Equations
• Physical Consequences of Complex Dimensions of Fractals
• Spectral analysis on infinite Sierpinski fractafolds
• Dirac and magnetic Schrödinger operators on fractals
• Existence of a Meromorphic Extension of Spectral Zeta Functions on Fractals
• Power-law energy level-spacing distributions in fractals
• Distance and tube zeta functions of fractals and arbitrary compact sets
• Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums
• Introduction to multifractional spacetimes
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
• Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis
• Singularly continuous spectrum of a self-similar Laplacian on the half-line
• Complex dimensions of fractals and meromorphic extensions of fractal zeta functions
• Wave equation on one-dimensional fractals with spectral decimation and the complex dynamics of polynomials
• Counting spanning trees on fractal graphs and their asymptotic complexity
• Fractal zeta functions and complex dimensions: A general higher-dimensional theory
• Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces
• Complex asymptotics of Poincaré functions and properties of Julia sets
• Spectral decimation of a self-similar version of almost Mathieu-type operators
• Regularized Laplacian determinants of self-similar fractals
• Geometry of fractional spaces
• Finite entanglement entropy and spectral dimension in quantum gravity
• Quantum Vacuum Energy of Self-Similar Configurations
• Disconnected Julia sets and gaps in the spectrum of Laplacians on symmetric finitely ramified fractals
• Counting Spanning Trees on Fractal Graphs
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets