Maps with dense orbits: Ansari's theorem revisited and the infinite torus

Abstract

Let $B$ be a Banach space and $T$ a bounded linear operator on $B.$ A celebrated theorem of Ansari says that whenever $T$ is hypercyclic so is any power $T^n$. We provide a very natural proof of this theorem by building on an approach by Bourdon. We also explore an extension to a non linear setting of a theorem of León-Saavedra and Müller which says that for $\lambda \in \mathbb C$ and $|\lambda|=1$ the operator $\lambda T$ is hypercyclic whenever $T$ is.