Arithmetic properties of families of plane polynomial automorphisms

TL;DR

This paper studies the arithmetic properties of periodic points in families of plane polynomial automorphisms, establishing bounds on their heights and finiteness results under certain conditions, extending previous work in the field.

Contribution

It generalizes known results by proving bounded height and finiteness of periodic parameters for families of Hénon maps over number fields, under mild conditions.

Findings

Bounded height of parameters with periodic fibers.

Finiteness of periodic parameters in non-periodic families.

Uniform bounds on the number of periodic points in non-degenerate families.

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.