Beyond half synchronized systems

In irreducible subshifts, a word m is synchronizing if whenever vm and mw are admissible words, then vmw is admissible as well. A word m is (left) half (resp. weak) synchronizing, when there is a left transitive ray (resp. a left ray) x− such that if x−m and mw are admissible, then x−mw is also admissible. The respective subshifts are called half (resp. weak) synchronized. K. Thomsen in [On the structure of a sofic shift space, American Mathematical Society, ۳۵۶, Number ۹, ۳۵۵۷-۳۶۱۹] considers a synchronized component of a general subshift and investigates the approximation of entropy from inside of this component by some certain SFTs. We, using a rather different approach, show how this result extends to weak synchronized systems.

Keywords:

half synchronized , Kreiger graph , Fischer cover , Entropy