We prove two relative local variational principles of topological pressure functions $P(T,\mathcal{ F},\mathcal{ U},y)$ and$P(T,\mathcal{ F},\mathcal{ U}|Y)$ for a given factor map $\pi$, an open cover $\mathcal{ U} $ and a subadditive sequence of real-valued continuous functions $\mathcal{ F}$. By proving the upper semi-continuity and affinity of the entropy maps $h_{ \{ \cdot\}}(T,\mathcal{ U}\mid Y)$ and $h^+_{ \{ \cdot\}}(T,\mathcal{ U}\mid Y)$ on the space of all invariant Borel probability measures, we show that the relative local pressure $P(T,\mathcal{ \{ \cdot\}},\mathcal{ U}|Y)$ for subadditive potentials determines the local measure-theoretic conditional entropies.
Categories: math.DS