Dr. Vasiliki Evdoridou (Universidad de Barcelona): "Non escaping endpoints of entire functions"
LMS4 - Raum 312
Let f_a= e^z+a, a<-1. The Julia set of f_a consists of an uncountable
union of disjoint curves going off to infinity (a Cantor bouquet).
Following several interesting results on the endpoints of these curves,
we consider the set of non-escaping endpoints, that is, the endpoints
whose iterates do not tend to infinity. We show that the union of
non-escaping endpoints with infinity is a totally separated set by
finding continua that separate these endpoints from infinity. This is a
complementary result to the very recent result of Alhabib and
Rempre-Gillen that for the same family of functions the set of escaping
endpoints together with infinity is connected. Moreover, we present
other functions in the exponential family as well as a function that was
first studied by Fatou which share the same property of the non-escaping
endpoints. Finally, we show how we can use the continua we have
constructed in order to show that the union of the Fatou set with the
set of points that escape to infinity 'as quickly as possible' has the
structure of a 'spider's web'.
Auf Einladung von Herrn Prof. Bergweiler.