On the variational principle for the topological pressure for certain non-compact sets

Let (X, d, T) be a dynamical system, where (X, d) is a compact metric space and T: X → X is a continuous map. We assume that the dynamical system satisfies g-almost product property and the uniform separation property. We compute the topological pressure of saturated sets under these two conditions. If the uniform separation property does not hold, we compute the topological pressure of the set of generic points. We give an application of these results to multifractal analysis and finally get a conditional variational principle.