Singular values and non-repelling cycles for entire transcendental maps

TL;DR

This paper investigates the dynamics of entire transcendental maps with bounded singular values, establishing a link between non-repelling cycles and singular orbits, and refining the Fatou-Shishikura inequality in finite singular value cases.

Contribution

It introduces a combinatorial approach to relate non-repelling cycles to singular orbits and refines existing inequalities for transcendental maps with finitely many singular values.

Findings

Each non-repelling cycle is associated with a unique singular orbit.

Singular orbits do not accumulate on other non-repelling cycles.

Refinement of the Fatou-Shishikura inequality for maps with finitely many singular values.

Abstract

Let $f$ be a map with bounded set of singular values for which periodic dynamic rays exist and land. We prove that each non-repelling cycle is associated to a singular orbit which cannot accumulate on any other non-repelling cycle. When $f$ has finitely many singular values this implies a refinement of the Fatou-Shishikura inequality. Our approach is combinatorial in the spirit of the approach used by [Ki00], [BCL+16] for polynomials.