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arXiv · 2607.06090

Transcendental correspondences: when Fuchsian groups take over basins of entire maps

Kostiantyn Drach·Sabyasachi Mukherjee·Leticia Pardo-Simón
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Abstract

In this paper, we initiate a systematic study of $(\infty : \infty)$ holomorphic correspondences that naturally arise as conformal combinations (matings) of transcendental entire maps with Fuchsian groups. This construction parallels the recent theory of finite-degree algebraic correspondences associated with rational maps. Our correspondence combines the dynamics of a transcendental entire function outside a distinguished attracting/parabolic basin with the action of a compatible Fuchsian group within it. We show that the resulting correspondence is the composition of a M\"obius involution and the deleted covering correspondence of a meromorphic function having exactly one simple pole. When the transcendental entire function has finitely many singular values, so does this meromorphic function, and its line complex can be described explicitly.

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Kostiantyn Drach, Sabyasachi Mukherjee, Leticia Pardo-Simón. 2026-07-07. Transcendental correspondences: when Fuchsian groups take over basins of entire maps. https://arxiv.org/abs/2607.06090

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