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Lyapunov exponents and related concepts for entire functions

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Abstract

Let f be an entire function and denote by \(f^\#\) the spherical derivative of f and by \(f^n\) the n-th iterate of f. For an open set U intersecting the Julia set J(f), we consider how fast \(\sup _{z\in U} (f^n)^\#(z)\) and \(\int _U (f^n)^\#(z)^2 dx\,dy\) tend to \(\infty \). We also study the growth rate of the sequence \((f^n)^\#(z)\) for \(z\in J(f)\).

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Acknowledgements

We thank Alexandre Eremenko, Dan Liu, Lasse Rempe-Gillen, Phil Rippon, Weixiao Shen and the referee for helpful comments.

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Authors and Affiliations

  1. Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Walter Bergweiler

  2. Shanghai Center for Mathematical Sciences, Shanghai, People’s Republic of China

    Xiao Yao

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, People’s Republic of China

    Jianhua Zheng

Authors
  1. Walter Bergweiler
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  2. Xiao Yao
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  3. Jianhua Zheng
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Corresponding author

Correspondence to Walter Bergweiler.

Additional information

X. Yao and J. Zheng were supported by the Grant (No. 11571193) of NSF of China.

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Bergweiler, W., Yao, X. & Zheng, J. Lyapunov exponents and related concepts for entire functions. Math. Z. 288, 855–873 (2018). https://doi.org/10.1007/s00209-017-1916-x

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  • DOI: https://doi.org/10.1007/s00209-017-1916-x

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