Topological Boundaries for Countable State Markov Shifts

Let ( S , σ) be a transitive locally compact countable state Markov shift. If ( Ŝ , σ ^ ) is some compact metric dynamical system (‘cmds’) that contains ( S , σ) as a dense subsystem then the space S * = Ŝ − S together with the homeomorphismσ * = σ ^ | S *will be called a boundary for ( S , σ). Any cmds ( X, T ) is shown to be a boundary for some transitive locally compact countable state Markov shift (Theorem A). We define a new canonical compactification whose boundary is a conjugacy invariant that distinguishes Markov shifts according to their behaviour ‘at infinity’ (Theorem B). This boundary, together with a periodic point condition, will be used to characterize the set of all chain‐recurrent boundaries of a given transitive locally compact countable state Markov shift (Theorems C, D).