Hypercyclic operators and rotated orbits with polynomial phases

TL;DR

This paper extends known results about hypercyclic operators to orbits rotated by polynomial phases, showing they remain hypercyclic under certain conditions and strengthening related non-linear results.

Contribution

It introduces new conditions under which rotated orbits of hypercyclic operators remain hypercyclic, especially with polynomial phases, and refines existing non-linear hypercyclicity results.

Findings

Rotations with polynomial phases preserve hypercyclicity.

Rapidly growing phases can destroy hypercyclicity.

Strengthens non-linear hypercyclicity results by Shkarin.

Abstract

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.