Abstract

For a Topological Dynamical System and a family of dynamical invariant subadditive continuous functions we introduce some new combinatorial and measure theoretical notions of pressure for a measurable cover conditioned with respect to a measurable partition following the same ideas used in [29]. Using the notion of conditional entropy defined in [29] we prove two local variational inequalities for this new notion of combinatorial conditional pressure. For the case of open covers with a clopen partition we prove that both inequalities reach equality for at least one common invariant measure. We also prove that for every invariant measure there is an equilibrium state generated by a single function that attains equality between this new conditional topological pressure and the measure theoretical conditional entropy. This is stated and proven in five Theorems. Finally we show that all these results stay true for the u.s.c. case. This extends the work done so far in the subadditive thermodynamic formalism for Z actions.