H\'enon maps with many rational periodic points

Building on work of Doyle and Hyde on polynomial maps in one variable, we produce for each odd integer $d \geq 2$ a H\'enon map of degree $d$ defined over $\mathbb{Q}$ with at least $(d-4)^2$ integral periodic points. This provides a quadratic lower bound on any conjectural uniform bound for periodic rational points of H\'enon maps. In contrast with the work of Doyle and Hyde, our examples also admit integer cycles of large period.