H\"older estimates and uniformity in arithmetic dynamics

Thomas Gauthier

H\"older estimates and uniformity in arithmetic dynamics

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arxiv 2407.13275 v2 pith:7RXDYOPB submitted 2024-07-18 math.DS math.NT

Thomas Gauthier This is my paper · ORCID

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keywords mapsrationalcommondegreesestimatesolderpointspreperiodic

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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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Heights and morphisms in number fields
math.NT 2024-11 conditional novelty 8.0 of 10

Provides an explicit power-saving asymptotic for counting points by pullback height under morphisms between projective spaces over number fields.

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