Hausdorff measure of hairs without endpoints in the exponential family

TL;DR

This paper investigates the Hausdorff measure of hairs without endpoints in the Julia sets of exponential maps, identifying conditions under which this measure is finite, thus deepening understanding of fractal geometry in complex dynamics.

Contribution

It determines for which gauge functions the Hausdorff measure of hairs without endpoints in exponential Julia sets is finite, extending previous dimension results.

Findings

Hausdorff dimension of hairs without endpoints is 1

Conditions on gauge functions for finite Hausdorff measure are established

Deepens understanding of fractal measures in complex dynamics

Abstract

Devaney and Krych showed that for $0<\lambda<1/e$ the Julia set of $\lambda e^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpi\'nska showed that the set of hairs without endpoints has Hausdorff dimension $1$. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.