On the algebraic properties of the automorphism groups of countable state Markov shifts

We study the algebraic properties of automorphism groups of two-sided, transitive, countable state Markov shifts together with the dynamics of those groups on the shiftspace itself as well as on periodic orbits and the 1-point-compactification of the shiftspace. We present a complete solution to the cardinality-question of the automorphism group for locally compact and non locally compact, countable state Markov shifts, shed some light on its huge subgroup structure and prove the analogue of Ryan’s theorem about the center of the automorphism group in the non-compact setting. Moreover we characterize the 1-point-compactification of locally compact, countable state Markov shifts, whose automorphism groups are countable and show that these compact dynamical systems are conjugate to synchronised systems on doubly-transitive points. Finally we prove the existence of a class of locally compact, countable state Markov shifts whose automorphism groups split into a direct sum of two groups; one being the infinite cyclic group generated by the shift map, the other being a countably infinite, centerless group, which contains all automorphisms that act on the orbit-complement of certain finite sets of symbols like the identity.