Quadratic Hénon Maps over Q with Rational Cycles of Length Three and Four for Every Jacobian
Year: 2026
Type: preprint
Abstract: In a list of open problems on Hénon maps (Arnold Math. J. 10 (2024)), P. Ingram conjectured that for all but finitely many δ ∈ Q every Q-rational periodic point of every map f(x, y) = (y, y² + c − δx) with c ∈ Q has period dividing 2. We show that the statement, as printed, is false for every δ ∈ Q, in each reading the text supports (finitely many exceptional δ; for fixed δ, finitely many exceptional c; finitely many exceptional pairs). For δ ≠ 1 and c = −(29δ² − 38δ + 29)/16 the point ((5 − 7δ)/4, −(1 + δ)/4) has period 3; for δ = 1 and c = −1 − s², s ∈ Q, there is a rational 3-cycle; for every δ ≠ 0 an explicit c gives a rational 4-cycle. Such cycles are essentially implicit in the source: the interpolation construction that opens the same section yields rational 4-cycles for every δ ∉ {0, 1}, and our 4-cycles are its normal form; so the printed conjecture contradicts its own section and is probably not the intended statement. We also describe all rational 3-cycles: shifted by 1/2, their x-coordinates are the rational points, off the line u = v = w, of the cone over the nodal cubic u³ + v³ + w³ − u²v − v²w − w²u = 0. Hence every δ ≠ 1 has infinitely many c with a rational 3-cycle, and for δ = 1 one exists if and only if −(1 + c) is a rational square. Computations, which are not theorems, find for δ = −1 exactly Ingram's periods 1, 2, 3, 4, 6, 8, and a rational 5-cycle for δ = −4 and, by a duality δ ↔ 1/δ, for δ = −1/4. The case δ = 0 is the classical map z² + c (Morton; Morton and Raianu), the case δ = −1 is Ingram's, and the novelty of the note is modest. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-152-0049 (Section 11 (P. Ingram), Conjecture 4 of "Hénon maps: a list of open problems", Arnold Math. J. 10 (2024) 585–620, arXiv:2312.03907).
Author: Ferudun, Alper
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222882520
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