A universal model for the bifurcations of asymptotic values
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910054823755776 |
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| author | Matthieu, Astorg Miriam, Benini Anna Núria, Fagella |
| author_facet | Matthieu, Astorg Miriam, Benini Anna Núria, Fagella |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_15466 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A universal model for the bifurcations of asymptotic values Matthieu, Astorg Miriam, Benini Anna Núria, Fagella Dynamical Systems Complex Variables 37F10, 37F12, 37F20, 37F25, 34F46 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. |
| title | A universal model for the bifurcations of asymptotic values |
| topic | Dynamical Systems Complex Variables 37F10, 37F12, 37F20, 37F25, 34F46 |
| url | https://arxiv.org/abs/2603.15466 |