Arithmetic properties of families of plane polynomial automorphisms

Abstract

Given an algebraic family $f\colon\Lambda\times \mathbb{A}^2 \to \Lambda\times \mathbb{A}^2$ of plane polynomial automorphisms of H\'enon type parameterized by a quasi-projective curve, defined over a number field $\mathbb{K},$ we investigate certain arithmetic properties of periodic points contained in a family of subvarieties $X \subset \Lambda \times \mathbb{A}^2 \twoheadrightarrow \Lambda$. First, consider $X$ as a curve. We prove that the set of parameters $t\in\Lambda(\overline{\mathbb{Q}})$, such that $X_t$ is periodic, has bounded height. This generalizes a result of Patrick Ingram. Moreover, if $X$ is non-periodic, then under some mild conditions -- such as when the family is dissipative -- we show that there are, in fact, only finitely many periodic parameters. This extends a result of Charles Favre and Romain Dujardin. Second, let $X$ be a family of curves. Assuming $X$ is non-degenerate, we establish a uniform bound on the number of periodic points in each curve $X_t$, $t\in \Lambda(\overline{\mathbb{Q}})$ and show that the set of these periodic points have bounded height in $\Lambda\times \mathbb{A}^2$ as well. We then examine in more detail the non-degeneracy property in the case of dissipative families of quadratic H\'enon maps.