{
  "schema_version": 1,
  "problem_number": "AMR-051-0010",
  "title": "Full-Dimensional One-Sided Trapping for Smooth Area-Preserving Plane Maps",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A point of an invertible plane dynamical system is one-sided trapped if its forward semiorbit is bounded and its backward semiorbit is unbounded. We construct a single smooth area-preserving diffeomorphism of the plane whose trapped set has Hausdorff dimension two. The map has the explicit form F(x,y)=(y,h(y)-x), with h given by a locally finite sum of smooth affine coding blocks. Its trapped set contains compact sets of dimensions 2-3/(j+3) for every positive integer j, and their backward orbits escape geometrically through a common channel. Thus the discrete-time maximum is two in every C^m class, including C-infinity. In contrast, for every complete jointly smooth autonomous area-preserving flow on the plane, a nonempty trapped set has dimension one. A self-contained critical-value argument proves this contrast. Finite-regularity continuous-time maxima are not determined. This is an unrefereed preprint; no independent expert review, formal verification or absolute priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.CA"
  ],
  "keywords": [
    "area-preserving diffeomorphism",
    "one-sided trapping",
    "Hausdorff dimension",
    "Cantor set",
    "Hamiltonian flow",
    "AMR-051-0010"
  ],
  "manuscript_version_date": "2026-10-17",
  "publication_date": "2026-10-17",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-17",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-051-0010/",
  "pdf_url": "https://eulersolve.org/papers/amr-051-0010/paper.pdf?v=4f8ac4465541",
  "doi": "10.5281/zenodo.23429529",
  "zenodo_record_url": "23429529",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The discrete-time maximum is two for every C^m class, attained by one C-infinity map. The continuous-time maximum is one for complete jointly C-infinity autonomous area-preserving plane flows. Finite-regularity continuous-time maxima remain unresolved. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
  "files": {
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      "sha256": "4f8ac44655419eb2ccad0245100bcb371692c620252e2545cac49b6a001afe0e"
    },
    "source.zip": {
      "sha256": "554d9ff02109cc3af19e97c8d9ea77ac21cadda6d691659731e6a0e218133bcf"
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    "verification_report.md": {
      "sha256": "6b147467f1ac3d2aba27395b30c525a4eeab9e74ce623b30cea5991748811c6a"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}