The successful applicant is expected to work on one of the four problems given below and is supervised by Núria Fagella and the supervisor within the area the applicant selects.
1 -- Boundness of Siegel disks
Supervisor: Núria Fagella (with possible collaboration of Christian Henriksen, team 7)
We consider a 1-(complex)parameter family of entire transcendental maps with a persistent Siegel disk,
one critical point and one asymptotic value. We propose a study of the parameter and dynamical planes to
study the relationship between the boundness of the Siegel disk and the behaviour of the asymptotic value.
This problem is related to the question of whether the Siegel disks of a.z.exp(z) are unbounded, for certain
values of a.
2 --Dynamics of quasiperiodically forced maps in the complex plane
Supervisor: Joan Carles Tatjer
The object of this project is to study an intermediate case between the dynamics of maps in the complex plane
and the dynamics in maps in spaces of complex dimension larger than one. The idea is to begin with simple cases,
comparing with the known results in real dynamics, and with the corresponding systems without forcing.
The role of the periodic points for the maps in the complex plane will be assumed by the closed invariant
sets in the quasiperiodically forced maps. This means that the concept of reducibility of invariant circles
will be relevant in this context.
3 -- Meromorphic functions with only one pole which is omitted
Supervisor: Xavier Jarque
This class of maps is, in the dynamical sense, in between entire and meromorphic functions. They share many
properties with both of the above classes. We propose a sistematic study of this class of functions from the
complex dynamics point of view.
4 -- A welding probleem
Supervisor: Joaquim Ortega-Cerdà
Recently Marshall and Rhode have proved the convergence of the so called zipper algorithm of numerical
conformal mapping. The proposed project consists in the implementation of a small variant of the algorithm
and an application of it to study a welding problem that arises when looking for drums D with a small inradius
(symply connected domains wich contain no big disks) and low fundamental frequency (the first Dirichlet
eigenvalue for the Laplacian in D). This domain should be a variant of the Goodman example as suggested in
the work by Bañuelos and Carroll. |
