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arXiv · 2603.15466

A universal model for the bifurcations of asymptotic values

Astorg Matthieu·Benini Anna Miriam·Fagella Núria
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Abstract

We study the notion of tangent-like maps, which is a transcendental analogue of polynomial-like maps. We introduce a model family analogous to quadratic polynomials, with only one free asymptotic value, and define the "Tandelbrot set" as the analogue of the Mandelbrot set. We prove a Straightening Theorem for tangent-like maps, with uniqueness of the model map in the case where the filled-in Julia set is connected, and a parameter version of the Straightening Theorem for suitable holomorphic families of tangent-like maps. As a consequence, we prove the existence of topological copies of the Tandelbrot set in bifurcation loci of numerous families of meromorphic maps.

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Astorg Matthieu, Benini Anna Miriam, Fagella Núria. 2026-03-16. A universal model for the bifurcations of asymptotic values. https://arxiv.org/abs/2603.15466

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