The order conjecture fails in S

TL;DR

This paper presents a counterexample in complex analysis, showing that the order of growth of certain entire functions can vary under quasiconformal equivalences, challenging a longstanding conjecture.

Contribution

It constructs a specific entire function with three singular values that disproves the Order Conjecture in the Speiser class ${ m S}$.

Findings

Counterexample to the Order Conjecture in ${ m S}$

Order of growth can change under quasiconformal equivalence

Demonstrates limitations of existing conjectures in complex dynamics

Abstract

We construct an entire function $f$ with only three singular values whose order of growth can change under a quasiconformal equivalence. This is a counterexample to the Order Conjecture in the Speiser class ${\mathcal S}$ of entire functions.