Estimate for supremum of conditional entropy on a closed subset (Q1012934)

The paper deals with topological dynamical system \((X,T),\) where \(X\) is a compact metric space and \(T\) is a continuous map from \(X\) to itself. Let \(G\) be closed fully \(T\)-invariant subspace of \(X\). The author proves that \[ h_{top}(T|G) \leq \sup_{\mu \in M(X,T)}h_\mu (T|\langle G\rangle) \leq h_{top}(T|G) + h_{top}(T|cl(X\setminus G)), \] where \(M(X,T)\) is the collection of all invariant measures \(\mu\) under \(T,\) \(cl(X\setminus G)\) is the closure of \(X\setminus G,\) \(h_{top}(T|G)\) is the topological entropy and \(h_{\mu}(T|\langle G\rangle)\) is the conditional metric entropy of \(T\) with respect to \(\mu\) and \(\langle G\rangle\).