Heights and periodic points for one-parameter families of H\'enon maps

TL;DR

This paper investigates the arithmetic and dynamical properties of a family of Hénon maps, focusing on height functions, local properties, and the distribution of periodic parameter values, including unlikely intersections.

Contribution

It establishes a connection between height functions of Hénon map families and semipositive adelically metrized line bundles, and analyzes the distribution of periodic parameters.

Findings

Height function $h_{{f P}}$ is linked to a semipositive adelic line bundle.

Characterization of when the set of periodic parameters $\Sigma({f P})$ is infinite.

Results on unlikely intersections of periodic parameter values.

Abstract

In this paper we study arithmetic properties of a one-parameter family ${\mathbf H}$ of H\'enon maps over the affine line. Given a family of initial points ${\mathbf P}$ satisfying a natural condition, we show the height function $h_{{\mathbf P}}$ associated to ${\mathbf H}$ and ${\mathbf P}$ is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We then show various local properties of $h_{{\mathbf P}}$. Next we consider the set $\Sigma({\mathbf P})$ consisting of periodic parameter values, and study when $\Sigma({\mathbf P})$ is an infinite set or not. We also study unlikely intersections of periodic parameter values.