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Poincaré Functional Equations, Harmonic Measures on Julia Sets, and Fractal Zeta Functions

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Fractal Geometry and Stochastics V

Part of the book series: Progress in Probability ((PRPR,volume 70))

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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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Authors and Affiliations

  1. Institut für Analysis und Computational Number Theory (Math A), NAWI-Graz, Technische Universität Graz, Steyrergasse 30, 8010, Graz, Austria

    Peter J. Grabner

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  1. Peter J. Grabner
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Correspondence to Peter J. Grabner .

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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany

    Christoph Bandt

  2. Mathematical Institute, University of St Andrews, St Andrews, United Kingdom

    Kenneth Falconer

  3. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Martina Zähle

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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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