# A bound on the number of rationally invisible repelling orbits

## TL;DR

This paper establishes a refined bound on the number of rationally invisible repelling orbits in certain entire transcendental maps, linking the count to the number of singular orbits and extending to maps with infinitely many singular values.

## Contribution

It introduces a new inequality that bounds rationally invisible repelling orbits based on the singular value structure, extending previous results to more general transcendental maps.

## Key findings

- Bound on rationally invisible repelling orbits is at most the number of singular orbits.
- The refined inequality applies to maps with finitely and infinitely many singular values.
- Techniques are applicable to broader classes of transcendental functions.

## Abstract

We consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land. For such maps, we prove a refined version of the Fatou-Shishikura inequality which takes into account rationally invisible periodic orbits, that is, repelling cycles which are not landing points of any periodic ray. More precisely, if there are $q<\infty$ singular orbits, then the sum of the number of attracting, parabolic, Siegel, Cremer or rationally invisible orbits is bounded above by $q$. In particular, there are at most $q$ rationally invisible repelling periodic orbits. The techniques presented here also apply to the more general setting in which the function is allowed to have infinitely many singular values.