About Entropy

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 resp ective subshifts are called half (resp. weak) synchronized. K. Thomsen considers a synchronized comp onent of a general subshift and investigates the approximation of entropy from inside of this comp onent by some certain SFTs. We, using a rather different approach, show how this result extends to weak synchronized systems. The notion of balanced generator of the Dyke system has b een extended to a general subshift called balanced shift and it has b een shown that they are essentially mixing. All balanced shifts are half synchronized and full shifts are the only balanced and synchronized subshifts. Also, a formula for the weak half synchronizing entropy of a balanced shift has b een given.