Let me consider the following space of sequences $\prod_{i\geq 1} \{0,1,2,3\}$ and define the so called shift map $$S:\prod_{i\geq 1} \{0,1,2,3\}\rightarrow \prod_{i\geq 1} \{0,1,2,3\};~~x:=(x_i)_{i\geq 1}\mapsto (x_{i+1})_{i\geq 1}$$
Our prof. remarked at some point that this map has periodic orbits (or more generally an arbitrary shift map has periodic orbits). To get some practice I wanted to find an explicit periodic orbit for $S$.
My idea was the following, let me pick for example $x=(0,3,2,1,0,3,2,1,0,3,2,1...)\in \prod_{i\geq 1} \{0,1,2,3\}$. Then we see that $$\begin{align}S(x)&=(3,2,1,0,3,2,1,0,3,2,1...)\\S^2(x)&=(2,1,0,3,2,1,0,3,2,1...)\\S^3(x)&=(1,0,3,2,1,0,3,2,1...)\\S^4(x)&=x\end{align}$$ so we see that $x$ has period $4$. Therefore in my opinion $\{x,S(x),S^2(x),S^3(x)\}$ is a periodic orbit.
Is this correct or am I wrong?