Inverses, powers and cartesian products of topologically deterministic maps

Abstract

We show that if $(X,T)$ is a topological dynamical system which is deterministic in the sense of Kamiński, Siemaszko and Szymański then $(X,T^{-1})$ and $(X\times X,T\times T)$ need not be deterministic in this sense. However if $(X\times X,T\times T)$ is deterministic then $(X,T^{n})$ is deterministic for all $n\in{\mathbb{N}}\setminus\{0\}$.