  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.
