Abstract
Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators. We propose an alternative notion of orbit forN-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclicN-linear operators, for eachN≥2. Indeed, the nonnormable spaces of entire functions and the countable product of lines supportN-linear operators with residual sets of hypercyclic vectors, forN=2.
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