Canonical heights for Henon maps

TL;DR

This paper studies the arithmetic properties of Henon maps over number and function fields, proving results about canonical heights, periodic points, and proposing a conjecture on rational periodic points with empirical verification.

Contribution

It establishes a variation result for Kawaguchi's canonical height in families of Henon maps and proves a bounded height set for parameters leading to periodic points.

Findings

Set of parameters with non-periodic points becoming periodic has bounded height.

Only points with canonical height zero are periodic in non-isotrivial cases.

Conjecture that rational periodic points for quadratic Henon maps have periods 1, 2, 3, 4, 6, or 8.

Abstract

We consider the arithmetic of Henon maps f(x, y)=(ay, x+f(y)) defined over number fields and function fields, usually with the restriction that a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Henon maps, and derive from this a specialization theorem, showing that the set of parameters above which a given non-periodic point becomes periodic is a set of bounded height. Proving this involves showing that the only points of canonical height zero for a Henon map over a function field are those which are periodic (in the non-isotrivial case). In the case of quadratic Henon maps f(x, y)=(y, x+y^2+b), we obtain a stronger result, bounding the canonical height below by a quantity which grows linearly in the height of b, once the number of places of bad reduction is fixed. Finally, we propose a conjecture regarding rational periodic points for quadratic Henon maps defined over the rational numbers, namely that they can only have period 1, 2, 3, 4, 6, or 8. We check this conjecture for the first million values of the parameter b, ordered by height.