Uniqueness of the equilibrium state in the dynamics of holomorphic correspondences

Bharath Krishna Seshadri; Shrihari Sridharan

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arXiv · 2607.28689

Uniqueness of the equilibrium state in the dynamics of holomorphic correspondences

Bharath Krishna Seshadri·Shrihari Sridharan

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Abstract

This paper concerns the study of the existence of a unique equilibrium state for a H\"{o}lder 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 H\"{o}lder 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.

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Bharath Krishna Seshadri, Shrihari Sridharan. 2026-07-30. Uniqueness of the equilibrium state in the dynamics of holomorphic correspondences. https://arxiv.org/abs/2607.28689

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