Let $X$ be a set, let $f: X \to X$ be a map, and let $A$ be a finite set. Let $\mathcal{P} = \{P_a : a \in A\}$ be a partition of $X$ such that for every $x \in X$ and every $n \in \mathbb{Z}_{\ge 0}$ there is a unique symbol $a_n(x) \in A$ with $f^n(x) \in P_{a_n(x)}$, where $f^0 = \operatorname{id}_X$ and $f^{n+1} = f \circ f^n$.

Define the itinerary map

\begin{align*} \iota: X \to A^{\mathbb{Z}_{\ge 0}} \end{align*}

by requiring that the $n$-th coordinate of $\iota(x)$ is $a_n(x)$ for every $n \in \mathbb{Z}_{\ge 0}$. Let

\begin{align*} \sigma: A^{\mathbb{Z}_{\ge 0}} \to A^{\mathbb{Z}_{\ge 0}} \end{align*}

be the one-sided left shift, defined by $(\sigma(s))_n = s_{n+1}$ for every sequence $s = (s_k)_{k \in \mathbb{Z}_{\ge 0}}$ and every $n \in \mathbb{Z}_{\ge 0}$. Then

\begin{align*} \iota \circ f = \sigma \circ \iota. \end{align*}