Abstract
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
$0<\kappa \leq 1/e$
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.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,General Mathematics
Cited by
1 articles.
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