A note on repelling periodic points for meromorphic functions with bounded set of singular values

TL;DR

This paper proves that meromorphic functions with bounded singular values and a logarithmic singularity at infinity have infinitely many repelling periodic points for all periods, using a simpler proof than previous methods.

Contribution

It introduces a simplified approach to show the existence of infinitely many repelling periodic points for such meromorphic functions.

Findings

Existence of infinitely many repelling periodic points for all periods n≥1.

Simpler proof technique compared to previous results for entire transcendental functions.

Applicable to meromorphic functions with bounded singular values and logarithmic singularity at infinity.

Abstract

Let $f$ be a meromorphic function with bounded set of singular values and for which infinity is a logarithmic singularity. Then we show that $f$ has infinitely many repelling periodic points for any minimal period $n\geq1$, using a much simpler argument than the corresponding results for arbitrary entire transcendental functions.