arXiv subjects

Fritz Vogl

Publications and source records attributed to Fritz Vogl.

3 recordsLinked to original sources

arXiv2012

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincaré and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpiński gasket.

math.SP↗

arXiv2007

Complex asymptotics of Poincaré functions and properties of Julia sets

The asymptotic behaviour of the solutions of Poincaré's functional equation $f(λz)=p(f(z))$ ($λ>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions of the complex plain. The constancy of an occurring periodic function is characterised in terms of geometric properties of the Julia set of $p$. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of $f$ is related to the harmonic measure on the Julia set of $p$.

math.CV↗

arXiv2005

The Zeta Function of the Laplacian on Certain Fractals

We prove that the zeta-function $ζ_Δ$ of the Laplacian $Δ$ on a self-similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane. We characterise the poles, compute their residues, and give expressions for some special values of the zeta-function. Furthermore, we discuss the presence of oscillations in the eigenvalue counting function.

math.SP↗