On the packing dimension of the Julia set and the escaping set of an entire function

TL;DR

This paper proves that for broad classes of entire functions, both the Julia set and the escaping set have packing dimension two, especially when the functions are bounded on certain curves or satisfy specific growth conditions.

Contribution

It establishes the packing dimension of Julia and escaping sets as two for large classes of entire functions under growth conditions.

Findings

Julia set and escaping set have packing dimension two for many entire functions.

The result applies when functions are bounded on curves tending to infinity.

Growth conditions involving minimum and maximum modulus ensure the dimension result.

Abstract

We show that for large classes of entire functions the Julia set and the escaping set have packing dimension two. For example, this is the case for entire functions which are bounded on a curve tending to infinity. More generally, we show that the result holds under suitable growth conditions involving the minimum and maximum modulus.