Let $A$ be a $0-1 \; N \times N$ matrix which has at least one $1$ in each row and each column. Let $\Omega_A$ be the set of all infinite sequences $\omega = (\omega_i)_{i \in \mathbb{Z}}, \omega_i \in \{0, ..., N-1\}$, such that $A_{\omega_i \omega_{i+1}} = 1, \forall i \in \mathbb{Z}$. If there is an element $\omega \in \Omega_A$ that contains the symbol $i$ at least twice then we call $i$ $\textit{essential}$.
I don't understand the definiton below:
Definition: We call two essential symbols $i$ and $j$ $\textit{equivalent}$ if there exists $\omega, \omega'\in \Omega_A, k_1 < k_2, l_1 <l_2$ such that $\omega_{k_1} = \omega_{l_2}' = i$ and $\omega_{k_2} = \omega_{l_1}' = j$, i. e. they occur in the same cycle.
Why does it mean that if $i$ and $j$ are equivalent essential symbols then they occur in the same cycle?
The infinite sequence $\omega$ passes through $i$ before $j$, whereas $\omega'$ passes through $j$ before $i$. Apparently we can walk in a cycle from $i$ to $j$ and back to $i$.