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Ergodicity, numerical range, and fixed points of holomorphic mappings

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Abstract

In this paper, we study the local structure of the fixed point set of a holomorphic mapping defined on a (not necessarily bounded or convex) domain in a complex Banach space, using ergodic theory of linear operators and the nonlinear numerical range introduced by L. A. Harris. We provide several constructions of holomorphic retractions and a generalization of Cartan’s Uniqueness Theorem. We also estimate the deviation of a holomorphic mapping from its linear approximation, the Fréchet derivative at a fixed point.

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Author information

Authors and Affiliations

  1. Department of Mathematics, The Technion — Israel Institute of Technology, 32000, Haifa, Israel

    Simeon Reich

  2. Department of Mathematics, Ort Braude College, 21982, Karmiel, Israel

    David Shoikhet

  3. Institute of Mathematics, Polish Academy of Sciences, P.O. Box 21, 00-956, Warszawa, Poland

    Jaroslav Zemánek

Authors
  1. Simeon Reich
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  2. David Shoikhet
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  3. Jaroslav Zemánek
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Corresponding author

Correspondence to Simeon Reich.

Additional information

S. Reich was partially supported by the Israel Science Foundation (Grant 647/07), by the Fund for the Promotion of Research at the Technion, and by the Technion President’s Research Fund.

D. Shoikhet was partially supported by the European Commission Project TODEQ (MTKD-CT-2005-030042).

J. Zemánek has been supported several times by ORT Braude College and by the European Commission Project TODEQ (MTKD-CT-2005-030042).

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Reich, S., Shoikhet, D. & Zemánek, J. Ergodicity, numerical range, and fixed points of holomorphic mappings. JAMA 119, 275–303 (2013). https://doi.org/10.1007/s11854-013-0009-y

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  • DOI: https://doi.org/10.1007/s11854-013-0009-y

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