# Symbolic dynamics

Symbolic dynamics in the narrow sense is the investigation of the topological Bernoulli automorphism $\sigma$ defined below, its invariant closed subsets, its invariant measures, etc. The topological Bernoulli automorphism $\sigma$ acts in the space $\Omega$ of infinite two-sided sequences of symbols from an alphabet $A$ (usually finite), provided with the direct product topology as an infinite number of copies of $A$ (it is usual to take the discrete topology in each copy). Namely, $\sigma$ takes a sequence $\omega = (\omega_i)$ to $\omega' = (\omega'_i)$, where $\omega_i' = \omega_{i+1}$ ( "shift the sequence one place to the left" ). An obvious generalization is the action of a group (or semi-group) $G$ on the space $A^G$ of functions from $G$ to $A$.
Let certain pairs $(a,b)$ of symbols from $A$ be declared "admissible" . All sequences $\omega_i$ for which for all $i$ the pair $(\omega_i,\omega_{i+1})$ is admissible form a closed ($\sigma$-) invariant subset $\Omega_1 \subset \Omega$. This is the most important example of an invariant subset of the topological Bernoulli automorphism. The dynamical system in $\Omega_1$ generated by the shift $\sigma{\downharpoonright}_{\Omega_1}$ is called a topological Markov chain.
Symbolic dynamics in the broad sense is the application of symbolic dynamics in the narrow sense to the investigation of dynamical systems which themselves are defined completely independently of $\Omega$ and $\sigma$.