The canonical-boundary representation for automorphism groups of locally compact countable state Markov shifts

We introduce the canonical-boundary representation and study its range. This conjugacy invariant homomorphism captures information about the symmetry of the Markov shift near its (canonical) boundary and exhibits which actions on the boundary can be realized by automorphisms. The path-structure at infinity — a relation on the set of orbits of the canonical boundary — is a new conjugacy invariant, which is stronger than the canonical boundary and the periodic data at infinity. Moreover we determine its influence on the range of the canonical-boundary representation and the extendability of automorphisms from subsystems (ascending sequences of shifts os finite type (SFTs) and infinite subsets of periodic points) to the entire Markov shift.