Synchronized entropy of totally synchronizing generated systems

We introduce the notion of a minimal generator \(G\) for a coded system \(X\); that is a generator \(G\) for \(X\) such that whenever \(u ∈ G\), then \(u ∈ W ({<G \{u\}>} )\). An \(X\) possessing such a \(G\) is called a minimally generated system. We introduce a class of minimally generated totally coded shift spaces generated by certain synchronizing blocks. For such shift spaces \(X\) we are able to show that if \(x ∈ X\), then there is unique \(\{… , v₋₁, v_0, v_1, v_2, … \} ⊂ G\) such that \(x = v₋₁v_0v_1v_2\). A carefully constructed example also shows that the converse of this statement is not necessarily true. The derived shift space \((∂ (X))\) of \(X\) plays an important role in the dynamics of the system. We characterize the derived shift space and use it to give a new shorter proof for computing synchronized entropy \(hsyn(X)\).