Compact factors of countable state Markov shifts

Abstract

We study continuous shift commuting maps from transitive countable state Markov shifts into compact subshifts. The closure of the image is a coded system. On the other hand, any coded system is the surjective image of some transitive Markov shift, which may be chosen locally compact by construction. These two results yield a formal analogy to “the transitive sofic systems are the subshift factors of transitive shifts of finite type”. Then we consider factor maps which have bounded coding length in some graph presentation (label maps). Now the image has to be synchronized, but not every synchronized system can be obtained in this way. We show various restrictions for a surjective label map to exist.