This paper establishes the geometric rigidity of certain holomorphic correspondences in the family $(w-c)^q=z^p$ , whose post-critical set is finite in any bounded domain of $\mathbb {C}$ . In spite of being rigid on the sphere, such correspondences are J-stable by means of holomorphic motions when viewed as maps of $\mathbb {C}^2$ . The key idea is the association of a conformal iterated function system to the return branches near the critical point, giving a global description of the post-critical set and proving the hyperbolicity of these correspondences.

中文翻译：

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双曲线对应的动力学

本文建立了 $(wc)^q=z^p$ 族中某些全纯对应的几何刚性，其后临界集在 $\mathbb {C}$ 的任何有界域中都是有限的。尽管在球面上是刚性的，但当被视为 $\mathbb {C}^2$ 的映射时，这种对应是通过全纯运动的J稳定的。关键思想是将保形迭代函数系统与临界点附近的返回分支相关联，给出后临界集的全局描述并证明这些对应关系的双曲线性。