An Infinite Family of Cubics with Emergent Reducibility at Depth 1

Abstract

A polynomial f(x) has emergent reducibility at depth n if f^{\circ k}(x) is irreducible for 0\leq k\leq n-1 but f^{\circ n}(x) is reducible. In this paper we prove that there are infinitely many irreducible cubics f \in \mathbb{Z}[x] with f\circ f reducible by exhibiting a one parameter family with this property.