Combinatorics and statistics for shifts of finite type

Consider all words over a finite alphabet that avoid a set of forbidden patterns, e.g. binary sequences with no two adjacent 1s. This can be viewed through the lens of ergodic theory, as a dynamical system (a 'shift space'); or combinatorial probability, as a Markov chain conditioned to avoid the forbidden set; or statistical physics, as the thermodynamic limit of a natural Gibbs measure. I will discuss connections between ideas from these worlds in the case where the forbidden set is of size one or two, including new results related to conjugacy of the underlying shifts, their entropies, and the asymptotic density of 1s.