H\'enon maps with many rational periodic points

TL;DR

This paper constructs specific Hénon maps over rationals with many rational periodic points, establishing a quadratic lower bound on the number of such points and demonstrating the existence of large-period integer cycles.

Contribution

It introduces a method to produce Hénon maps with a quadratic number of rational periodic points, expanding understanding of periodic dynamics in polynomial automorphisms.

Findings

Constructed Hénon maps with at least (d-4)^2 rational periodic points

Provided quadratic lower bounds on uniform bounds for rational periodic points

Demonstrated existence of integer cycles with large periods

Abstract

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.