A local variational principle for conditional entropy

For a given factor map $\pi:X\longrightarrow Y$ between two topological dynamical systems and a Borel cover ${\mathcal U}$, two notions of measure-theoretical conditional entropy $h_\mu^+(T,{\mathcal U}\mid Y)$ and $h_\mu^-(T,{\mathcal U}\mid Y)$ for an invariant Borel probability measure $\mu$ are introduced. It is shown that $h_\mu^+(T,{\mathcal U}\mid Y)=h_\mu^-(T,{\mathcal U}\mid Y)$. Moreover, $\max_{\mu}h_\mu^+(T,{\mathcal U}\mid Y)=h_{{\rm top}}(T,{\mathcal U}\mid Y)$ when $\mathcal U$ is an open cover. The relative variational principle is a consequence of the results.