Consider the shift space: $\Omega_N := \{\omega = (\omega_i)_{i \in \mathbb{Z}}: \omega_i \in \{0, ..., N-1\}, \; \forall \; i \in \mathbb{Z} \}$ and the shift map $\sigma : \Omega_N \to \Omega_N$, $\sigma(\omega)_i = \omega_{i+1}$. If $k \in \mathbb{N}$ then, a $\textit{block of length k}$ is an element of the set $\Omega_N^k = \Omega_N \times ... \times \Omega_N$ ($k$ times). If $\alpha \in \Omega_N^k$, then $\alpha = \alpha_0 ... \alpha_{k-1}$.
A non-empty block $\alpha$ is said to occur in $\omega \in \Omega_N \; \text{at place} \; i$ whenever $$\alpha = \omega_i ...\omega_{i + |\alpha| -1},$$ where $|\alpha|$ is the length of $\alpha$.
$\omega \in \Omega_N$ is called $\textit{positively recurrent}$ if $\exists \; n_k \to \infty $ such that $\omega = \underset{k \to \infty}{lim} \sigma^{n_k}(\omega)$.
Can someone give me a hint on how to show that $$\omega \; \textit{is positively recurrent iff every block which occurs in} \; \omega \; \textit{does so at places i, for arbitrarily large i} \; ?$$
Thank you!
It follows from the notion of convergence that $\sigma^{n_k}(\omega)$ converges to $\omega$ if and only if for each $m\ge1$ the sequences $\omega$ and $\sigma^{n_k}(\omega)$ have the same first $m$ components for all sufficiently large $k$. But this is possible if and only if $$ \text{every block which occurs in $\omega$ does so at places $i$, for arbitrarily large $i$}. $$ Incidentally, better rephrasing the later for example as $$ \text{every block in $\omega$ occurs infinitely often}. $$