Variational principles of topological pressure for correspondences

Recently, Li, Li and Zhang introduced the topological pressure for correspondences and measure-theoretic entropy for transition probability kernels. Building thereon, they established a variational principle for correspondences satisfying the forward expansiveness condition. In this work, we extend this research by deriving two types of variational principles: (i) For a class of correspondences, the topological pressure equals the supremum of the measure-theoretic pressures over extreme points of invariant measures. (ii) An abstract variational principle holds for general correspondences without requiring forward expansiveness. Furthermore, the differentiability and equilibrium states of the topological pressure for correspondences are also investigated.