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Prove that there exists an $n\in\mathbb{Z}\cup\left\{-\infty,+\infty\right\}$ such that… (Dynamics)

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and on it the following dynamics described by $T\colon X\to X$ as follows: A 1 becomes a 2, a 2 becomes a 0 and a 0 becomes a 1 if at least one of its two ...