On the formulas of meromorphic functions with periodic Herman rings (Q2089699)

In this well-written paper, the author constructs some explicit formulas for rational maps and transcendental meromorphic functions having Herman rings of period strictly larger than one, thus answering a question raised by \textit{M. Shishikura} [Ann. Sci. Éc. Norm. Supér. (4) 20, No. 1, 1--29 (1987; Zbl 0621.58030)] in the 1980s. For the construction, the author first performs the quasiconformal surgery which transforms the cycles of Siegel disks to (non-nested) Herman rings, and then applies the deformation theory in the Herman rings to get the desired formulas in the main theorems (Theorems B and C). In Theorem D, the author also gives the explicit formula of a family of rational maps having a 2-cycle of nested Herman rings. Furthermore, the author shows that the maps in Theorem B can also have a 2-cycle of interlaced Herman rings. In the final section, the author proves Proposition 3.3, which gives a family of transcendental entire functions having Siegel disks of any period and any Brjuno rotation number. This is a key ingredient for performing the quasiconformal surgery in Theorem C. The paper contains many illuminating pictures.