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 sofc shift space, American Mathematical Society, 356, Number 9, p. 557-3619] considers a synchronized component of a general subshift and investigates the approximation of entropy from inside of this component by some certain SFT’s. We, using a rather different approach, show how this result extends to weak synchronized systems.