A faithful symbolic extension
In this paper \((X,T)\) is a topological dynamical system and \((Y,S)\) is a symbolic extension of it, i.e. \((X,T)\) is a factor of a symbolic system \((Y,S)\) and there is a continuous mapping \(\phi : Y\rightarrow X\) such that \(\phi \circ S=T \circ \phi\). The author also introduces the extension entropy function of an extension \(\phi : Y\rightarrow X\) by setting \(h^{\phi}_{\text{ext}}(\mu)=\sup\{h(S,\nu)\}\) where \(\nu\) is a Borel probability measure on \(X\) invariant under \(T\), and \(\phi_{*}(\nu)=\nu \circ \phi^{-1}=\mu\). With these notions the author proves the following interesting theorem. If \((X,T)\) is a finite entropy, zero-dimensional dynamical system without periodic points and if \(\psi ~:~(Y,S)\rightarrow (X,T)\)is a symbolic extension, then there is another symbolic extension \(\phi~:~(W,S)\rightarrow (X,T)\) such that: (i) \(h^{\phi}_{\text{ext}}=h^{\psi}_{\text{ext}}\) on \(M(X,T)\); (ii) \(\phi_{*}~:~M(W,S)\rightarrow M(X,T)\) is invertible.
scientific article; zbMATH DE number 6159049