An important result of Léon-Saavedra and Müller 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 Léon-Saavedra and Müller's result in a nonlinear setting.
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