2025/12/17 by Zhang, Yu, Zhu, Yujun
Computer Science · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Topological and Geometric Data Analysis
paper · doi:10.48550/arxiv.2512.15238
openalex publication_date 2025/12/17 · openalex created_date 2025/12/19 · openalex updated_date 2026/07/28
In \citeMiller-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 \citeXiaoran 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 \citeXiaoran 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 C2-expanding endomorphisms. When the generators of T have no coincidence point, a locally Hölder continuous potential ϕ of two variables is defined via the Jacobians of the generators. The pressure of ϕ and its equilibrium state (μ, Q) are obtained, where μ is a T-invariant measure which is absolutely continuous with respect to the volume and Q is the associated transition probability kernel satisfying μQ=μ. 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.