Relative local variational principles for subadditive potentials

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