Interpolation of power mappings

@article{Burkart2022InterpolationOP,
  title={Interpolation of power mappings},
  author={Jack Burkart and Kirill Lazebnik},
  journal={Revista Matem{\'a}tica Iberoamericana},
  year={2022}
}
Let (Mj)j=1 ∈ N and (rj)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 : rj · exp(π/Mj) ≤ |z| ≤ rj+1} is given by a perturbed rescaling of z 7→ zj , such that the only singular values of f are rescalings of ±rj j . We describe several applications to the dynamics of entire functions. 

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