Nowhere differentiable hairs for entire maps

TL;DR

This paper constructs an entire function whose Julia set consists of hairs that are nowhere differentiable, contrasting previous results where hairs were smooth or differentiable.

Contribution

It introduces a novel entire function with Julia set hairs that are nowhere differentiable, expanding understanding of the geometric complexity of Julia sets.

Findings

Julia set hairs can be nowhere differentiable

Contrasts with previous smooth or differentiable hairs results

Provides new example of complex geometric structures in dynamics

Abstract

In 1984 Devaney and Krych showed that for the exponential family $\lambda e^z$, where $0<\lambda <1/e$, the Julia set consists of uncountably many pairwise disjoint simple curves tending to $\infty$, which they called hairs. Viana proved that these hairs are smooth. Bara\'nski as well as Rottenfusser, R\"uckert, Rempe and Schleicher gave analogues of the result of Devaney and Krych for more general classes of functions. In contrast to Viana's result we construct in this article an entire function, where the Julia set consists of hairs, which are nowhere differentiable.