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 $f: \mathbb{ C} \rightarrow\mathbb{ C}$ whose behavior in the large annuli $\{ z\in\mathbb{ C} : r_{ j}\cdot\exp(\pi/M_{ j})\leq|z|\leq r_{ j+1}\}$ 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.
Categories: math.CV math.DS