
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…
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