{
  "abstract": "We study the Longest Edge Bisection (LEB) process as a dynamical system on the projective shape space of simplices. A long-standing conjecture going back to Adler and Rivara-Levin and motivated by finite-element mesh refinement, often taken as a standing assumption, is that this procedure is non-degenerate and, in fact, in a certain way periodic. We prove: • There are 3-dimensional simplices such that the longest edge-bisection algorithm degenerates. • There is an open set of 4-dimensional simplices on which the longest edge-bisection algorithm degenerates. • If parametrizing the space of d-dimensional simplices by independent standard Gaussian vectors, then as d increases, a random simplex degenerates asymptotically almost surely. This is realized through exhibiting hyperbolic behaviour of the LEB process. We also exhibit elliptic behaviour that is nonperiodic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08846",
  "author_total": 3,
  "authors": [
    {
      "name": "Karim Adiprasito"
    },
    {
      "name": "Daniel Kalmanovich"
    },
    {
      "name": "Yaar Solomon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fvde2gjj0z3e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08846",
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  "id": "3fvde2gjj0z3e",
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  "landing_url": "https://hal.science/hal-05742991v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
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      "1704|Computational Geometry and Mesh Generation"
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  "pdf_url": "https://hal.science/hal-05742991/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fvde2gjj0z3e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:31Z",
      "native_id": "hal-05742991",
      "source": "hal",
      "url": "https://hal.science/hal-05742991v1"
    },
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08846",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08846"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-mg",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Degenerating orbits of the Longest Edge Bisection process",
  "title_text": "Degenerating orbits of the Longest Edge Bisection process",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/3fvde2gjj0z3e",
  "venue": "HAL",
  "versions": []
}