Explosion points and topology of Julia sets of Zorich maps

TL;DR

This paper introduces a topological model for Julia sets of Zorich maps, demonstrating that infinity acts as an explosion point for endpoints and exploring the embedding properties of associated hairy surfaces.

Contribution

It presents a novel topological model for Zorich map Julia sets and analyzes their embedding properties, extending understanding of higher-dimensional exponential dynamics.

Findings

Infinity is an explosion point for endpoints of Julia sets.

Introduces the concept of hairy surfaces as compactified Julia sets.

Shows non-uniqueness of embedding hairy surfaces in 3^3.

Abstract

Zorich maps are higher dimensional analogues of the complex exponential map. For the exponential family $\lambda e^z$, $\lambda>0$, it is known that for small values of $\lambda$ the Julia set is an uncountable collection of disjoint curves. The same was shown to hold for Zorich maps by Bergweiler and Nicks. In this paper we introduce a topological model for the Julia sets of certain Zorich maps, similar to the so called \textit{straight brush} of Aarts and Oversteegen. As a corollary we show that $\infty$ is an \textit{explosion point} for the set of endpoints of the Julia sets. Moreover we introduce an object called a \textit{hairy surface} which is a compactified version of the Julia set of Zorich maps and we show that those objects are not uniquely embedded in $\mathbb{R}^3$, unlike the corresponding two dimensional objects which are all ambiently homeomorphic.