Interpolation of Power Mappings

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}\c...

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Bibliographic Details
Published in:arXiv.org 2022-06
Main Authors: Burkart, Jack, Lazebnik, Kirill
Format: Article
Language:English
Subjects:
Entire functions
Interpolation
Rescaling
Online Access:Get full text
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Summary: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.
ISSN:2331-8422