No Herman rings for regularly ramified rational maps

Hu, Jun; Xiao, Yingqing

doi:10.1090/proc/14347

No Herman rings for regularly ramified rational maps
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by Jun Hu and Yingqing Xiao

Proc. Amer. Math. Soc. 147 (2019), 1587-1596

DOI: https://doi.org/10.1090/proc/14347

Published electronically: January 9, 2019

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Abstract:

It is due to Milnor that there is no Herman ring for any rational map with two critical values. In this paper, we are concerned with if there exist Herman rings for rational maps with three critical values. We first prove that such rational maps cannot have Herman rings of period $1$, $2$, or $3$. Then we prove all regularly ramified rational maps have no Herman rings in their Fatou sets.

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