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arXiv · 2407.13275

H\"older estimates and uniformity in arithmetic dynamics

Thomas Gauthier
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Abstract

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 H\"older properties of the Green function of a rational map to obtain height estimates.

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Thomas Gauthier. 2024-07-18. H\"older estimates and uniformity in arithmetic dynamics. https://arxiv.org/abs/2407.13275

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