  We present a number of rigidity results concerning holomorphic dynamical
systems admitting rotation quasicircles. Firstly, we show the absence of line
fields on the Julia set of any rational map that is geometrically finite away
from a number of rotation quasicircles with bounded type rotation number. As an
application, we prove combinatorial rigidity associated to the problem of
degeneration of Herman rings of the simplest configuration. Secondly, we extend
a result of de Faria and de Melo on the $C^{1+\alpha}$ rigidity of critical
circle maps with bounded type rotation number to a larger class of dynamical
objects, namely critical quasicircle maps. Unlike critical circle maps,
critical quasicircle maps may have imbalanced inner and outer criticalities. As
a consequence, we prove dynamical universality and exponential convergence of
renormalization towards a horseshoe attractor.
