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

TL;DR

This paper establishes a uniform bound on the number of common periodic points for families of regular plane polynomial automorphisms, extending previous results from rational maps to polynomial automorphisms over algebraic curves.

Contribution

It proves a uniform bound on common periodic points or the existence of a shared iterate for families of polynomial automorphisms, generalizing prior work to a new setting.

Findings

Either the automorphisms share a common iterate or the number of common periodic points is bounded by a constant.

The result applies to families parameterized by algebraic curves over number fields.

Extension of Mavraki and Schmidt's result from rational maps to 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.