{
  "abstract": "We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product $S^2\\times S^2$ with two fixed points. The argument starts with a four-fixed-point map, passes to rescaled cotangent coordinates near the antidiagonal, and reduces the local fixed-point equations to two planar saddle problems. A relative modification merges each pair of saddles, and mixed generating functions are used to transfer the modification to the original map. We include detailed explanations of the normal form along a level arc, the Moser equation, uniform existence of mixed generating functions, the order of parameter choices, and the global graph argument. This construction gives a counterexample to the degenerate version of Arnold's conjecture proposed by Arnold\\cite{Arnold86} when the manifold is standard $S^2\\times S^2$. This work is done with ChatGPT. The author gave the idea to study the limit behavior of $Φ_ρ$. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33626",
  "author_total": 1,
  "authors": [
    {
      "name": "Hao Jiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9wx82fy5a3t7y.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33626",
  "fulltext_url": null,
  "id": "9wx82fy5a3t7y",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33626",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214762721",
  "pdf_url": "https://arxiv.org/pdf/2609.33626",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9wx82fy5a3t7y.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2609.33626",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33626"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on $S^2\\times S^2$",
  "title_text": "A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on S^2× S^2",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/9wx82fy5a3t7y",
  "venue": "arXiv",
  "versions": []
}