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A remark on Schröder’s equation: formal and analytic linearization of iterative roots of the power series \(f(z)=z\)

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Abstract

We study Schröder’s equation (i.e. the problem of linearization) for local analytic functions \(F\) with \(F(0)=0, F'(0) \ne 1, F'(0)\) a root of \(1\). While Schröder’s equation in this case need not have even a formal solution, we show that if \(F\) is formally linearizable, then it can also be linearized by an invertible local analytic transformation. On the other hand, there exist also divergent series solutions of Schröder’s equation in this situation. We give some applications of our results to iterative functional equations, functional-differential equations and iteration groups.

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Acknowledgments

The partial support of the internal research project F1R-MTH-PUL-12RDO2 of the University of Luxembourg is gratefully acknowledged.

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Authors and Affiliations

  1. Institute of Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010,  Graz, Austria

    Ludwig Reich

  2. Mathematics Research Unit, Université du Luxembourg, 6, rue Richard Coudenhove-Kalergi, 1359,  Luxembourg, Grand Duchy of Luxembourg

    Jörg Tomaschek

Authors
  1. Ludwig Reich
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  2. Jörg Tomaschek
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Correspondence to Jörg Tomaschek.

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Communicated by A. Constantin.

J. Tomaschek is supported by the National Research Fund, Luxembourg (AFR 3979497), and cofunded under the Marie Curie Actions of the European Commission (FP7-COFUND)

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Reich, L., Tomaschek, J. A remark on Schröder’s equation: formal and analytic linearization of iterative roots of the power series \(f(z)=z\) . Monatsh Math 175, 411–427 (2014). https://doi.org/10.1007/s00605-013-0601-3

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