Prescribing the Postsingular Dynamics of Meromorphic Functions

TL;DR

This paper demonstrates that any discrete planar sequence's postsingular dynamics can be realized by a transcendental meromorphic function with small perturbations, extending rational dynamics results to the transcendental setting.

Contribution

It introduces a method to construct meromorphic functions with prescribed postsingular dynamics and geometric control, generalizing finite sequence results to infinite sequences.

Findings

Realized postsingular dynamics for any discrete planar sequence

Developed a construction method using the Folding Theorem and Tychonoff's fixpoint theorem

Extended rational dynamics results to transcendental meromorphic functions

Abstract

We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of $S$. This work was influenced by an analogous result of DeMarco, Koch and McMullen for finite $S$ in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of $f$ and the geometry of $f$, using the Folding Theorem of Bishop and a classical fixpoint theorem of Tychonoff.