AI 中文总结
本文针对黎曼球面上的全纯对应动力学，证明了Dinh-Sibony测度支撑集上Hölder连续势函数的平衡态唯一性，还建立了Ruelle算子与平衡态的关联并给出相关示例。
AI 中文摘要
本文研究黎曼球面上定义的全纯对应动力学下，Hölder连续函数的唯一平衡态的存在性。我们主要在Dinh-Sibony测度的支撑集上分析该对应，识别出拓扑性质良好的对应，即距离扩张对应。进一步，我们考虑定义在Dinh-Sibony测度支撑集上的Hölder连续势函数，并证明其平衡态的唯一性。在此过程中，我们还证明了与全纯对应相关的若干有趣拓扑结果。最后，我们建立了一个结果，在合适假设下将全纯对应的Ruelle算子与唯一平衡态关联起来。论文的结论部分讨论了所涉及的假设并提供了若干示例。
英文摘要
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.
Comments21 pages. Comments are welcome!
