Topological Pressure of Discontinuous Potentials of Flow and Variational Principle

Let X be a compact metric space and Φ=\φₜ\ₜ be a continuous flow on X. We introduce two types of topological pressure for family of discontinuous potentials a=ₜ>0. First, define the topological pressure of family of measurable potentials a=ₜ>0 on a subset Z for flow and proof its invariant principle. The second topological pressure is defined on a invariant subset having a nested family of subsets, we also proof its invariant principle.