Universal elements for non-linear operators and their applications

Abstract

We prove that under certain topological conditions on the set of universal elements of a continuous map $T$ acting on a topological space $X$, that the direct sum $T\oplus M_g$ is universal, where $M_g$ is multiplication by a generating element of a compact topological group. We use this result to characterize ${\mathbb{R}}_{+}$-supercyclic operators and to show that whenever $T$ is a supercyclic operator and $z_1,...,z_n$ are pairwise different non-zero complex numbers, then the operator $z_{1}T\oplus ...\oplus z_{n}T$ is cyclic. The latter answers affirmatively a question of Bayart and Matheron.