ESTIMATE FOR SUPREMUM OF CONDITIONAL ENTROPY ON A CLOSED SUBSET

Abstract

This paper compares the conditional metric entropy $h_{\mu}(T\mid G)$, with the topological entropy, $h_{top}(T\mid G)$, of a continuous map $T$, where $G$ is a closed fully $T$-invariant subset. The following Variational Inequality is proven, $$h_{top}(T\mid G)\leq \sup _{\mu\in M(X,T)}h_{\mu}(T\mid \langle G \rangle )\leq h_{top}(T\mid G)+h_{top}(T\mid cl(X\backslash G))$$ where $M(X,T)$ is the collection of all invariant measures of $X$, which is an extension of the usual variational principle when $G=X$.