2026/07/30 by Bharath Krishna Seshadri, Shrihari Sridharan
Mathematics · #math.DS #msc:37A35 #msc:37A60 #msc:37F80
21 pages. Comments are welcome!
arxiv created 2026/07/30 · arxiv updated 2026/08/03
This paper concerns the study of the existence of a unique equilibrium state for a Hölder continuous function under the dynamics of a holomorphic correspondence defined on the Riemann sphere. We mainly work with the correspondence restricted on the support of the Dinh-Sibony measure and identify topologically interesting correspondences, namely distance expanding ones. Further, we consider Hölder continuous potentials defined on the support of the Dinh-Sibony measure, for which we prove the uniqueness of equilibrium state. Along the way, we also prove some interesting topological results related to holomorphic correspondences. Finally, we establish a result connecting the Ruelle operator for holomorphic correspondences and the unique equilibrium state under a suitable hypothesis. The concluding part of the paper is devoted to some discussion related to the hypothesis involved and providing some examples.