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arXiv · 1711.02365

Dynamics of hyperbolic correspondences

Carlos Siqueira
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Abstract

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. We also show that Julia sets of any perturbation of such correspondences are obtained as $\alpha$ limit sets of typical points, establishing the hyperbolicity of these correspondences.

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Carlos Siqueira. 2017-11-07. Dynamics of hyperbolic correspondences. https://doi.org/10.1017/etds.2021.49

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