Uniform bound on common periodic points for families of regular plane polynomial automorphisms

Abstract

Given two one-dimensional families $f$ and $g$ of regular plane polynomial automorphisms parameterised by an algebraic curve $B$, all defined over some number field $K$, such that one of them is dissipative, we prove that at any parameter $b\in B(\mathbb{C})$, either $f_b$ and $g_b$ share a common iterate, or the number of their common periodic points $\mathrm{Per}(f_b)\cap \mathrm{Per}(g_b)$ is bounded by a uniform constant $D$ (independent of the parameter $b$). We thus extend a result of Mavraki and Schmidt for rational maps to our setting.