A variational principle for discontinuous potentials

Let X be a compact space, f : X -> X a continuous map, and Lambda subset of X be any f-invariant subset. Assume that there exists a nested family of subsets {Lambda(l)}(l >= 1) that exhaust Lambda, that is Lambda(l) subset of A(l+1) and Lambda = boolean OR(l >= 1) Lambda(l). Assume that the potential phi: X -> R is continuous on the closure of each At but not necessarily continuous on Lambda. We define the topological pressure of phi on Lambda. This definition is shown to have a corresponding variational principle. We apply the topological pressure and variational principle to systems with nonzero Lyapunov exponents, countable Markov shifts, and unimodal maps.