Dimensions of orbital sets in complex dynamics

TL;DR

This paper investigates the relationship between the box dimensions of orbital sets, Julia sets, and initial sets in complex dynamics, extending inhomogeneous iterated function system results to rational maps.

Contribution

It introduces new dimension estimates for orbital sets in complex dynamics, generalizing previous results from iterated function systems.

Findings

Upper box dimension of orbital sets relates to Julia set dimensions

Results extend inhomogeneous IFS theory to complex rational maps

Provides bounds connecting initial set and Julia set dimensions

Abstract

Let $ E $ be a non-empty compact subset of the Riemann sphere and $T$ be a rational map of degree at least two. We study the associated \emph{orbital set}, that is, the backwards orbit of $E$ under $T$, and study the relationship between the upper box dimension of the orbital set and the upper box dimensions of the Julia set of $T$ and the initial set $ E$. Our results extend previous work on inhomogeneous iterated function systems to the setting of complex dynamical systems.