Hölder estimates and uniformity in arithmetic dynamics
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915031989354496 |
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| author | Gauthier, Thomas |
| author_facet | Gauthier, Thomas |
| contents | In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees $d_1,d_2\geq2$, outside a Zariski closed subset of the space of pairs of rational maps $(f,g)$ of degree $d_1$ and $d_2$ respectively, the maps $f$ and $g$ share at most a uniformly bounded number of common preperiodic points. This generalizes a result of DeMarco and Mavraki to maps of possibly different degrees. Our main contribution is the use of Hölder properties of the Green function of a rational map to obtain height estimates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13275 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hölder estimates and uniformity in arithmetic dynamics Gauthier, Thomas Dynamical Systems Number Theory In this note we study common preperiodic points of rational maps of the Riemann Sphere. We show that given any degrees $d_1,d_2\geq2$, outside a Zariski closed subset of the space of pairs of rational maps $(f,g)$ of degree $d_1$ and $d_2$ respectively, the maps $f$ and $g$ share at most a uniformly bounded number of common preperiodic points. This generalizes a result of DeMarco and Mavraki to maps of possibly different degrees. Our main contribution is the use of Hölder properties of the Green function of a rational map to obtain height estimates. |
| title | Hölder estimates and uniformity in arithmetic dynamics |
| topic | Dynamical Systems Number Theory |
| url | https://arxiv.org/abs/2407.13275 |