Interpolation of power mappings

. Let ( M j ) ∞ j =1 ∈ N and ( r j ) ∞ j =1 ∈ R + be increasing sequences satisfying some mild rate of growth conditions. We prove that there is an entire function f : C → C whose behavior in the large annuli { z ∈ C : r j · exp( π/M j ) ≤ | z | ≤ r j +1 } is given by a perturbed rescaling of z (cid:55)→ z M j , such that the only singular values of f are rescalings of ± r M j j . We describe several applications to the dynamics of entire functions.