Julia sets of Zorich maps

TL;DR

This paper extends the understanding of Julia sets from exponential maps to Zorich maps in three dimensions, showing they can be the entire space, with dense periodic points and connected escaping sets.

Contribution

It generalizes classical results about Julia sets and dynamics from exponential maps to Zorich maps in three dimensions, including the entire space Julia set and dense periodic points.

Findings

Julia set of certain Zorich maps is the entire ^3.

Periodic points are dense in ^3.

Escaping set of Zorich maps is connected.

Abstract

The Julia set of the exponential family $E_{\kappa}:z\mapsto\kappa e^z$, $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa\leq1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the entire $\mathbb{R}^3$ generalizing Misiurewicz's result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb{R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.