Topological pressure and equilibrium state for certain correspondences

In \cite{Miller-Akin1999}, Miller and Akin investigated the invariant measures for correspondences, which are also known as upper semi-continuous set-valued maps. Recently, the variational principle and thermodynamic formalism for forward expansive correspondences were studied by Li, Li and Zhang \cite{Xiaoran Li-Zhiqiang Li-Yiwei Zhang2023}. In this paper, the invariant measures and the associated transition probability kernels are explicitly expressed for certain correspondences satisfying the assumptions in \cite{Xiaoran Li-Zhiqiang Li-Yiwei Zhang2023} via the equilibrium states of some particular potentials. Let $T$ be a correspondence on a closed connected Riemannian manifold generated by finite $C^{2}$-expanding endomorphisms. When the generators of $T$ have no coincidence point, a locally H\"{o}lder continuous potential $\phi$ of two variables is defined via the Jacobians of the generators. The pressure of $\phi$ and its equilibrium state $(\mu, \mathcal{Q})$ are obtained, where $\mu$ is a $T$-invariant measure which is absolutely continuous with respect to the volume and $\mathcal{Q}$ is the associated transition probability kernel satisfying $\mu\mathcal{Q}=\mu$. For the correspondence $T$ on the torus whose generators have coincidence points, the variational topological pressures for measurable potentials are introduced and the corresponding equilibrium states are considered. Moreover, the uniqueness of the equilibrium states of correspondences is considered via the natural extensions.