Countable state Markov shifts with automorphism groups being a direct sum

Abstract

In this short note we prove the existence of a class of transitive, 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 $H$, which contains all automorphisms that act on the orbit-complement of certain finite sets of symbols like the identity. Such a decomposition is well known from the automorphism groups of coded systems, in which case one can explicitly construct example subshifts with $\aut(\sigma)=\seq{\sigma}\oplus H$ to a variety of abstract groups $H$. A similar result for shifts of finite type (SFTs) is yet only established for full $p$-shifts ($p$ prime), where $H$ equals the set of inert automorphisms. For general SFTs no direct sum representation is known so far. Thus our result may help to distinguish between the countable automorphism groups of SFTs and those of countable state Markov shifts.