  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.
