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H\'enon maps with many rational periodic points

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Hyeonggeun Kim, Holly Krieger, Mara-Ioana Postolache, Vivian Szeto

Dynamical Systems · Submitted 2 Dec 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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arXiv ID2412.01668 (version 2)
Pith IDGPAMVNQW, Pith's permanent record of this paper and its review
CitationHolly Krieger, Hyeonggeun Kim, Mara-Ioana Postolache, Vivian Szeto. "H\'enon maps with many rational periodic points." arXiv:2412.01668 (2024). https://pith.science/paper/GPAMVNQW
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,968 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.97 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

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.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 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. result 0.56 0.96 0.87 asserted 1.00
s1 This provides a quadratic lower bound on any conjectural uniform bound for periodic rational points of H\'enon maps. interpretation 0.99 0.67 0.29 asserted 0.82
s2 In contrast with the work of Doyle and Hyde, our examples also admit integer cycles of large period. result 0.75 0.89 0.08 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original 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.

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