Julia sets of Zorich maps

Athanasios Tsantaris

doi:10.1017/etds.2021.123

Abstract

AbstractThe 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 $0<\kappa \leq 1/e$ 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 whole of $\mathbb {R}^3$ , 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.

Highlights

In the study of the dynamics of complex analytic functions one of the most well-studied and important families of functions is the exponential family Eκ : z → κez, κ ∈ C − {0}
On the other hand, when κ > 1/e, Misiurewicz in [23] proved that the Julia set J(Eλ) equals the entire complex plane C ( Misiurewicz only proved this for κ = 1 but his proof can be adapted to cover the other cases as well; see [10])
Following [18], we describe the construction of the Zorich maps in three dimensions

Summary

Introduction

In the study of the dynamics of complex analytic functions one of the most well-studied and important families of functions is the exponential family Eκ : z → κez, κ ∈ C − {0}. Perhaps the most fundamental fact about this family concerns its Julia set. The Julia set J(f ) of an entire function f is the set of all points in the complex plane where the family of iterates f n of f is not normal. On the other hand, when κ > 1/e, Misiurewicz in [23] proved that the Julia set J(Eλ) equals the entire complex plane C ( Misiurewicz only proved this for κ = 1 but his proof can be adapted to cover the other cases as well; see [10]). For a different proof of the same fact, see [29] For all these facts and much more we refer to Devaney’s survey [11] on exponential dynamics

Tsantaris

Escaping set of the Zorich maps