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}
}
• Published 11 January 2021
• Mathematics
• Revista Matemática Iberoamericana
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.
1 Citations

Figures from this paper

Transcendental Julia Sets of Minimal Hausdorff Dimension
• Mathematics
• 2021
We show the existence of transcendental entire functions f : C → C with Hausdorff-dimension 1 Julia sets, such that every Fatou component of f has infinite inner connectivity. We also show that there

References

SHOWING 1-10 OF 50 REFERENCES
Multiply connected wandering domains of entire functions
• Mathematics
• 2013
The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function f
Entire functions of slow growth whose Julia set coincides with the plane
• Mathematics
Ergodic Theory and Dynamical Systems
• 2000
We construct a transcendental entire function $f$ with $J(f)=\mathbb{C}$ such that $f$ has arbitrarily slow growth; that is, $\log |f(z)|\leq\phi(|z|)\log |z|$ for $|z|>r_0$, where $\phi$ is an
An entire function which has wandering domains
• I. Baker
• Mathematics
Journal of the Australian Mathematical Society
• 1976
Abstract Let f(z) denote a rational or entire function of the complex variable z and fn(z), n = 1,2, …, the n−th iterate of f. Provided f is not rational of order 0 or 1, the set of those points
Prescribing the postsingular dynamics of meromorphic functions
• Mathematics
Mathematische Annalen
• 2019
We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of
A transcendental Julia set of dimension 1
We construct a non-polynomial entire function whose Julia set has finite 1-dimensional spherical measure, and hence Hausdorff dimension 1. In 1975, Baker proved the dimension of such a Julia set must
On the set where the iterates of an entire function are neither escaping nor bounded
• Mathematics
• 2015
For a transcendental entire function f, we study the set of points BU(f) whose iterates under f neither escape to infinity nor are bounded. We give new results on the connectedness properties of this
Speiser class Julia sets with dimension near one
• Mathematics
Journal d'Analyse Mathématique
• 2020
For any $\delta >0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=\delta$. Stallard proved that the dimension must be strictly
The Teichmuller Space of an Entire Function
• Mathematics
• 2008
We consider the Teichmuller space of a general entire transcendental function f : C → C regardless of the nature of the set of singular values of f (critical values and asymptotic values). We prove
Arc-like continua, Julia sets of entire functions, and Eremenko's Conjecture
A hyperbolic transcendental entire function with connected Fatou set is said to be "of disjoint type". It is known that a disjoint-type function provides a model for the dynamics near infinity of all