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 oil the shift space itself as well as on periodic orbits and the 1-point-compactification of the shift space. We present a complete Solution to the cardinality question of the automorphism a-locally compact. countable-state Markov shifts, shed group for locally compact and non 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-cornpactification of locally compact, countable-state Markov shifts, whose automorphism groups are countable and show that these compact dynamical systems are conjugate to synchronized 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.