Canonical heights for Hénon maps

Abstract

We consider the arithmetic of Hénon maps φ(x,y)=(ay,x+f(y)) defined over number fields and function fields, usually with the restriction a=1. We prove a result on the variation of Kawaguchi's canonical height in families of Hénon maps, and derive from this 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 Hénon map over a function field are those which are periodic (in the non‐isotrivial case). In the case of quadratic Hénon maps φ(x, y)=(y, x+y2+b), we obtain a stronger result, bounding the canonical height from below by a quantity that grows linearly in the height of b, once the number of places of bad reduction is fixed. Finally, we propose a conjecture regarding ℚ‐rational periodic points for quadratic Hénon maps defined over ℚ, namely that they can only have period 1, 2, 3, 4, 6, or 8. We check this conjecture for the first million values of b∈ℚ, ordered by height.

Variation of the canonical height for a family of polynomials

“…Generalizing work of Ghioca and Mavraki [12], Mavraki and Ye [23] established the same for rational functions in the case X = B = P 1 , but under the additional hypothesis that the pair (f, P ) is quasi-adelic, a condition which is not known to always hold. Related results also exist for Hénon maps [14], Drinfeld modules [15], and dynamical correspondences [17]. As well, the aforementioned result of Tate has been sharpened by DeMarco and Mavraki [6] to show thatĥ Et (P t ) is precisely a height induced by an adelically metrized line bundle on the base, while the case of single-variables polynomials from [13] was subsequently similarly strengthened by Favre and Gauthier [9].…”

Residual periodicity on the Markoff Surface

“…The first fact can be shown for example by using bounds on valuations as in Ingram [12]. The second fact reduces to example (7) by looking at the diagonal x = y.…”

Canonical heights for correspondences

“…where h U is any height on U with respect to a divisor class of degree 1, and o(1) → 0 as h U (t) → ∞ (see [11,14,15,16] for improvements and variations in some special cases).…”