Abstract
We collect results from earlier work with G. Derfel and F. Vogl, which led to the proof of the existence of a meromorphic continuation of the fractal zeta function for certain self-similar fractals admitting spectral decimation. We explain the connection to classical functional equations occurring in the theory of polynomial iteration, namely Poincaré’s and Böttcher’s equations, as well as properties of the harmonic measure on the underlying Julia set. Furthermore, we comment on some more recent developments based on the work of N. Kajino and state a conjecture related to our approach via functional equations.
The author is supported by the Austrian Science Fund (FWF): projects F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”), W1230 (Doctoral Program “Discrete Mathematics”), and I1136 (French-Austrian international cooperation “Fractals and Numeration”).
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Grabner, P.J. (2015). Poincaré Functional Equations, Harmonic Measures on Julia Sets, and Fractal Zeta Functions. In: Bandt, C., Falconer, K., Zähle, M. (eds) Fractal Geometry and Stochastics V. Progress in Probability, vol 70. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18660-3_10
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