  An important result of Le\'on-Saavedra and M\"uller says that the rotations
of hypercyclic operators remain hypercyclic. We provide extensions of this
result for orbits of operators which are rotated by unimodular complex numbers
with polynomial phases. On the other hand, we show that this fails for
unimodular complex numbers whose phases grow to infinity too quickly, say at a
geometric rate. A further consequence of our work is a notable strengthening of
a result due to Shkarin which concerns variants of Le\'on-Saavedra and
M\"uller's result in a non-linear setting.
