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A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on S2× S2

  • Hao Jiao
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Abstract

We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product S2× S2 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 S2× S2. This work is done with ChatGPT. The author gave the idea to study the limit behavior of Φρ. GPT suggested a key idea of using second generating function to construct this Hamiltonian. Also, most of the computation is done by GPT.

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Cite

BibTeXRIS
BibTeX
@misc{jiao2026smooth,
  title = {{A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on $S^2\times S^2$}},
  author = {Hao Jiao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.33626},
  eprint = {2609.33626},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33626},
}
RIS
TY  - UNPB
TI  - A Smooth Hamiltonian Diffeomorphism with Two Fixed Points on S^2× S^2
AU  - Hao Jiao
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.33626
UR  - https://arxiv.org/abs/2609.33626
AB  - We give a construction of a Hamiltonian diffeomorphism of the equal-area symplectic product S^2× 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× 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.
ER  -