Prescribing the postsingular dynamics of meromorphic functions

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 is motivated by an analogous result of DeMarco et al. (Mathematische Annalen, 2018) 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 (Acta Math 214(1):1–60, 2015) and a classical fixpoint theorem (Tychonoff in Math Ann 111(1):767–776, 1935).