Rafael de la Llave, Nikola P. Petrov
Boundaries of Siegel disks - numerical studies of their dynamics and regularity
(1347K, pdf)
ABSTRACT. Siegel disks are domains around fixed points of holomorphic maps
in which the maps are locally linearizable
(i.e., become a rotation under an appropriate change of
coordinates which is analytic in a neighborhood of the origin).
The dynamical behavior of the iterates of the map
on the boundary of the Siegel disk
exhibits strong scaling properties
which have been intensively studied
in the physical and mathematical literature.
In the cases we study,
the boundary of the Siegel disk is a Jordan curve
containing a critical point of the map
(we consider critical maps of different orders),
and there exists a natural parameterization
which transforms the dynamics on the boundary into a rotation.
We compute numerically this parameterization and
use methods of harmonic analysis
to compute the global Holder regularity
of the parameterization
for different maps and rotation numbers.
We obtain that the regularity of the boundaries
and the scaling exponents
are universal numbers in the sense of renormalization theory
(i.e., they do not depend on the map when the map
ranges in an open set),
and only depend on the order of the critical point
of the map in the boundary of the Siegel disk
and the tail of the continued function expansion
of the rotation number.
We also discuss some possible relations
between the regularity of the parameterization
of the boundaries and the corresponding
scaling exponents.