Interpolation of Power Mappings

Abstract

Let $(M_j{)}_{j=1}^{\infty }\in \mathbb{N}$ and $(r_j{)}_{j=1}^{\infty }\in {\mathbb{R}}^{+}$ be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function $f:\mathbb{C}\to \mathbb{C}$ whose behavior in the large annuli $\{z\in \mathbb{C}:r_j\cdot \exp (\pi /M_j)\le |z|\le r_{j+1}\}$ is given by a perturbed rescaling of $z\mapsto z^{M_j}$, such that the only singular values of $f$ are rescalings of $\pm r_j^{M_j}$. We describe several applications to the dynamics of entire functions.