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On the formulas of meromorphic functions with periodic Herman rings

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Abstract

We construct some explicit formulas of rational maps and transcendental meromorphic functions having Herman rings of period strictly larger than one. This gives an answer to a question raised by Shishikura in the 1980s. Moreover, the formulas of some rational maps with nested Herman rings are also found. To obtain the formulas of transcendental meromorphic functions having periodic Herman rings, a crucial step is to find an explicit family of transcendental entire functions having bounded Siegel disks of all possible periods and rotation numbers. This is based on proving the existence of a Mandelbrot-like set of period one in the corresponding parameter space.

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Acknowledgements

The author would like to thank Arnaud Chéritat for providing an algorithm to draw Fig. 4 and Lasse Rempe for very helpful comments. He is also very grateful to the referee for very insightful and detailed comments, suggestions and corrections. This work was supported by NSFC (Grant No. 12071210) and NSF of Jiangsu Province (Grant No. BK20191246).

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Fei Yang

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The author states that there is no conflict of interest. The datasets generated during the current study are available from the author on reasonable request.

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Yang, F. On the formulas of meromorphic functions with periodic Herman rings. Math. Ann. 384, 989–1015 (2022). https://doi.org/10.1007/s00208-021-02308-1

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