On the formulas of meromorphic functions with periodic Herman rings

TL;DR

This paper constructs explicit formulas for rational and transcendental meromorphic functions with Herman rings of various periods, answering a question from the 1980s and expanding understanding of complex dynamics.

Contribution

It provides the first explicit formulas for functions with Herman rings of period greater than one and nested Herman rings, advancing the field of complex dynamics.

Findings

Explicit formulas for rational maps with Herman rings of period > 1

Explicit formulas for transcendental meromorphic functions with Herman rings

Existence of a Mandelbrot-like set for functions with bounded Siegel disks

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.