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Spectral Zeta Function of Symmetric Fractals

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Fractal Geometry and Stochastics III

Part of the book series: Progress in Probability ((PRPR,volume 57))

Abstract

This is an expository paper which includes several topics related to the spectral analysis on fractals. Such analysis on certain symmetric fractals can be completely described in terms of complex dynamics of a polynomial. The examples of such fractals include the Sierpiński gasket and an interval with a fractal measure. We discuss the spectral type of the Laplacian, complex spectral dimensions and spectral zeta function. The spectral zeta function has a product structure that involves a certain new zeta function of a polynomial and a “geometric” part. A similar product structure in the case of fractal strings was discovered by M. L. Lapidus. We give examples were the spectrum is singularly continuous on one dimensional fractals but is pure point on the infinite Sierpiński gasket, with a complete set of compactly supported eigenfunctions. We describe the spectrum of the Laplacian (in terms of the complex dynamics of a polynomial) and all the eigenfunctions with compact support.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Alexander Teplyaev

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  1. Alexander Teplyaev
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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Ernst-Moritz-Arndt-Universität, Greifswald, 17487, Germany

    Christoph Bandt

  2. Department of Physics, University of Rome La Sapienza, Via G. Boni 20, Roma, 00162, Italy

    Umberto Mosco

  3. Mathematisches Institut, Friedrich-Schiller-Universität, Jena, 07740, Germany

    Martina Zähle

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Teplyaev, A. (2004). Spectral Zeta Function of Symmetric Fractals. In: Bandt, C., Mosco, U., Zähle, M. (eds) Fractal Geometry and Stochastics III. Progress in Probability, vol 57. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7891-3_16

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  • DOI: https://doi.org/10.1007/978-3-0348-7891-3_16

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9612-2

  • Online ISBN: 978-3-0348-7891-3

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