On postsingularly finite exponential maps

TL;DR

This paper investigates the distribution of parameters in exponential maps where zero is preperiodic, providing asymptotic counts of such parameters within large disks in the complex plane.

Contribution

It determines the asymptotic behavior of the number of parameters with preperiodic zero under exponential maps, for given preperiod and period.

Findings

Asymptotic growth rate of parameter count as radius increases

Explicit formulas for parameter distribution in exponential maps

Insights into the structure of postsingularly finite exponential maps

Abstract

We consider parameters $\lambda$ for which $0$ is preperiodic under the map $z\mapsto\lambda e^z$. Given $k$ and $l$, let $n(r)$ be the number of $\lambda$ satisfying $0<|\lambda|\leq r$ such that $0$ is mapped after $k$ iterations to a periodic point of period $l$. We determine the asymptotic behavior of $n(r)$ as $r$ tends to $\infty$.