Given a number field $\mathbb{ K} \subset \mathbb{ C}$ that is not contained in $\mathbb{ R}$, we prove the existence of a dense set of entire maps $f \colon \mathbb{ C} \rightarrow \mathbb{ C}$ whose preperiodic points and multipliers all lie in $\mathbb{ K}$. This contrasts with the case of rational maps. In addition, we show that there exists an escaping quadratic-like map that is not conjugate to an affine escaping quadratic-like map and whose multipliers all lie in $\mathbb{ Q}$.
Categories: math.DS