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

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 $\mathrm{\Sigma }(\mathbf{P})$ consisting of periodic parameter values, and study when $\mathrm{\Sigma }(\mathbf{P})$ is an infinite set or not. We also study unlikely intersections of periodic parameter values.