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Power Spectrum and Fredholm Determinant Related to Intermittent Chaos

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Stochastic Processes in Physics and Engineering

Part of the book series: Mathematics and Its Applications ((MAIA,volume 42))

Abstract

We present a mathematical aspect to the problem of intermittent chaos. The singularity of power spectra, such as l/ω-spectrum, is shown for a class of correlation functions under the dynamics called semi-Bernoulli systems and is also proved to be common among this class. The singularity is described by a power series called the Fredholm determinant, which may be regarded as det(I-zL) for the transfer operator L although L has a strange spectral property: every complex number in the open unit disk is an eigenvalue. Finally examples are given among maps of intervals and among statistical mechanics of one dimensional lattice models to conclude that the l/ω-spectrum is a critical phenomenon at the tri-critical point.

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Author information

Authors and Affiliations

  1. Dept. of Pure and Applied Sciences, University of Tokyo, Komaba, Meguro, Tokyo 153, Japan

    Y. Takahashi

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  1. Y. Takahashi
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Editor information

Editors and Affiliations

  1. Fachbereich Mathematik, Ruhr-Universität Bochum, Postfach 10 21 48, 4630, Bochum, Germany

    Sergio Albeverio

  2. Fakultät für Physik, Universilät Bielefeld, Postfach 86 40, 4800, Bielefeld 1, Germany

    Philip Blanchard  & Ludwig Streit  & 

  3. Centrum voor Wiskunde en Informatica, Kruislaan 413, Postbus 4079, 1098 SJ, Amsterdam, The Netherlands

    Michiel Hazewinkel

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© 1988 D. Reidel Publishing Company

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Takahashi, Y. (1988). Power Spectrum and Fredholm Determinant Related to Intermittent Chaos. In: Albeverio, S., Blanchard, P., Hazewinkel, M., Streit, L. (eds) Stochastic Processes in Physics and Engineering. Mathematics and Its Applications, vol 42. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-2893-0_19

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  • DOI: https://doi.org/10.1007/978-94-009-2893-0_19

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-7803-0

  • Online ISBN: 978-94-009-2893-0

  • eBook Packages: Springer Book Archive

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