### 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 $n\in{\mathbb{N}}\setminus\{0\}$.

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 $n\in{\mathbb{N}}\setminus\{0\}$.

#### Keywords

Topological dynamics; topological determinism; recurrence

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