A new geometric characterization of the Julia set

TL;DR

This paper introduces a novel geometric approach to characterize Julia sets using Ahlfors-Shimizu's characteristic, providing growth estimates that deepen understanding of their structure.

Contribution

It presents a new geometric characterization of Julia sets based on growth estimates involving Ahlfors-Shimizu's characteristic, advancing the theoretical understanding of complex dynamics.

Findings

Established growth results for Julia sets using Ahlfors-Shimizu's characteristic

Provided bounds on the lower limit of $S(f^n,U)$ for neighborhoods in Julia sets

Enhanced the geometric understanding of Julia sets through new characterization methods

Abstract

This article concerns a new geometric characterization of the Julia set. By using Ahlfors-Shimizu's characteristic, we establish some growth results which indicates the characterization of the Julia set. The main technique is to estimate the lower bound of $S(f^n,U)$, where $U$ is an open neighbourhood of some point in $\mathcal{J}(f)$.