Synchronized entropy of totally synchronizing generated systems

Abstract

We introduce the notion of a minimal generator \(G\) for a coded system \(X\); that is a generator \(G\) for \(X\) such that whenever \(u \in G\), then \(u \not \in W \big(\overline{<G \setminus \{u\}>} \big)\). 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 \(x \in X\), then there is unique \(\{\ldots , v_{-1}, v_0, v_1, v_2, \ldots \} \subset G\) such that \(x = v_{-1}v_0v_1v_2\). A carefully constructed example also shows that the converse of this statement is not necessarily true.

The derived shift space \((\partial (X))\) 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 \(h_{syn}(X)\).