Ultrafilters in Arithmetic Dynamics

Given a one-parameter degenerating family of rational maps on the projective line, it is possible to construct a non-archimedean limit which captures how this family degenerates.  Recently, Luo used ultrafilters to construct limits for an arbitrary sequence of rational maps. This has been recently formalized by Favre-Gong using Berkovich spaces. In this talk, I will explain how to use Favre-Gong's theory to obtain (ineffective) uniformity results in arithmetic dynamics. In particular, I will prove that the number of common preperiodic points of two distinct monic degree d polynomials is uniformly bounded in terms of d, proving a conjecture of DeMarco-Krieger-Ye for monic polynomials.