# Explosion points and topology of Julia sets of Zorich maps

@inproceedings{Tsantaris2021ExplosionPA, title={Explosion points and topology of Julia sets of Zorich maps}, author={Athanasios Tsantaris}, year={2021} }

Zorich maps are higher dimensional analogues of the complex exponential map. For the exponential family λez , λ > 0, it is known that for small values of λ 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 straight brush of Aarts and Oversteegen. As a corollary we show that ∞ is an explosion point for…

## 2 Citations

Julia sets of Zorich maps

- MathematicsErgodic Theory and Dynamical Systems
- 2021

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…

Hausdorff dimension in quasiregular dynamics

- Mathematics
- 2022

. It is shown that the Hausdorﬀ dimension of the fast escaping set of a quasiregular self-map of R 3 can take any value in the interval [1 , 3]. The Hausdorﬀ dimension of the Julia set of such a map…

## References

SHOWING 1-10 OF 37 REFERENCES

An explosion point for the set of endpoints of the Julia set of λ exp (z)

- MathematicsErgodic Theory and Dynamical Systems
- 1990

Abstract The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to…

The geometry of Julia sets

- Mathematics
- 1993

The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on…

Julia sets of Zorich maps

- MathematicsErgodic Theory and Dynamical Systems
- 2021

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…

A characterization of smooth Cantor bouquets

- Mathematics
- 1990

We prove that all smooth fans having a dense set of endpoints are topologically equivalent. Let X be a smooth fan whose set of endpoints is dense in X. Such fans have been constructed, e.g., by J. H.…

Foundations for an iteration theory of entire quasiregular maps

- Mathematics
- 2014

The Fatou-Julia iteration theory of rational functions has been extended to uniformly quasiregular mappings in higher dimension by various authors, and recently some results of Fatou-Julia type have…

Non‐escaping endpoints do not explode

- MathematicsBulletin of the London Mathematical Society
- 2018

The family of exponential maps fa(z)=ez+a is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set J(fa) .…

Dynamics of generalised exponential maps

- Mathematics
- 2019

Since 1984, many authors have studied the dynamics of maps of the form
$\mathcal{E}_a(z) = e^z - a$
, with
$a > 1$
. It is now well-known that the Julia set of such a map has…

Fatou-Julia Theory for non-uniformly quasiregular maps

- Mathematics
- 2011

Montel’s theorem plays a central role in the Fatou-Julia theory of iteration of rational functions. Montel’s theorem has been extended to quasiregular maps in higher dimensions by Rickman and…