An extension of hypercyclicity for \(N\)-linear operators (Q1724519)

Summary: Grosse-Erdmann and Kim recently introduced the notion of bihypercyclicity for studying the existence of dense orbits under bilinear operators [\textit{K.-G. Grosse-Erdmann} and \textit{S. G. Kim}, J. Math. Anal. Appl. 399, No. 2, 701--708 (2013; Zbl 1272.47009)]. We propose an alternative notion of orbit for \(N\)-linear operators that is inspired by difference equations. Under this new notion, every separable infinite dimensional Fréchet space supports supercyclic \(N\)-linear operators, for each \(N \geq 2\). Indeed, the nonnormable spaces of entire functions and the countable product of lines support \(N\)-linear operators with residual sets of hypercyclic vectors, for \(N = 2\).