Hausdorff and Lesbesgue measure of Julia sets (task D3)
While it was shown only recently by Buff and Cheritat that there are
polynomials with Julia sets of positive measure, the corresponding
result for transcendental entire functions was proved already in
the 1980s by Eremenko and Lyubich and by McMullen, who showed that
this is the case for trigonometric functions and who also considered
the Hausdorff dimension of the Julia sets of exponential functions.
Since then a lot of work has been devoted to the Hausdorff dimension
and area of Julia sets. Most of this work (with notable exceptions
e.g. by Stallard) has been devoted to functions where the set of
singular values is finite or bounded. The main tool used for functions
with bounded set of singularities is a logarithmic change of variables
introduced to the subject by Eremenko and Lyubich. The task would
be to
obtain results about Hausdorff dimension and area of Julia sets for
certain classes of functions with unbounded set of singularities. Of
course, here the logarithmic change of variables has to be replaced by
other arguments. A condition one could pose is that the set of
singularities is sufficiently "thin". Under a suitable hypothesis of
this type Bargmann could prove that there are no invariant Baker
domains (a result normally requiring the logarithmic change of
variables). It seems possible that under hypotheses of this type one
can also say something about the Hausdorff dimension and area of
Julia sets. Other conditions that play a role here are suitable
notions of hyperbolicity. Questions concerning the area of Julia
sets are closely connected to ergodicity, and such problems are also
part of the research task.
The task will be performed in Kiel (Germany) by a postdoc appointed
as ER for 12 months. There will be regular contact in particular with
Bremen and Göttingen and with the British, the Polish, and the
Spanish team.
|
