Interpolation of Power Mappings

TL;DR

This paper constructs entire functions with controlled growth and singular values, using interpolation techniques based on sequences, and explores their applications in complex dynamics.

Contribution

It introduces a method to interpolate entire functions with prescribed behavior and singular values, advancing the understanding of entire function dynamics.

Findings

Existence of entire functions with specified growth in annuli

Controlled placement of singular values as rescaled points

Applications demonstrated in complex dynamics scenarios

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 $f: \mathbb{C} \rightarrow\mathbb{C}$ 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.