H\"older estimates and uniformity in arithmetic dynamics

TL;DR

This paper investigates the intersection of preperiodic points of rational maps on the Riemann Sphere, establishing uniform bounds outside a Zariski closed set by leveraging H"older continuity of Green functions.

Contribution

It generalizes previous results to rational maps of different degrees using H"older estimates of Green functions for height bounds.

Findings

Uniform bound on common preperiodic points for generic pairs of rational maps.

Extension of DeMarco and Mavraki's results to maps with different degrees.

Application of H"older properties of Green functions to arithmetic dynamics.

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.