{
  "abstract": "We study the similarity classes of matrices in $U(1,1;\\mathbb H)$ and the iterates of their associated quaternionic Möbius transformations. Building on the known elliptic case, we show that every parabolic or loxodromic element of $U(1,1;\\mathbb H)$ is similar, via a matrix in $U(1,1;\\mathbb H)$, to a complex matrix. Explicit complex representatives and matrices realizing these similarities are constructed for the different types. We then introduce the invariants $\\operatorname{Tr}(T), Δ(T), η(T), κ(T)$, and prove that they form a complete system of similarity invariants for matrices in $U(1,1;\\mathbb H)$. Finally, using the complex representation of quaternionic matrices and Chebyshev polynomials, we derive a fourth-order recurrence for a real sequence $(q_n)$ satisfying $Δ(T^n)=q_n^2Δ(T)$. Together with recurrence relations for the relevant matrix entries, this identity yields a complete classification of the quaternionic Möbius transformation associated with $T^n$, including the degenerate cases in which $Δ(T^n)=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24580",
  "author_total": 3,
  "authors": [
    {
      "name": "Bingzhe Hou"
    },
    {
      "name": "Cailing Yao"
    },
    {
      "name": "Jiaxing He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bnxhtcwyr7mc7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24580",
  "fulltext_url": null,
  "id": "bnxhtcwyr7mc7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24580",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Algebraic and Geometric Analysis"
    ]
  ],
  "openalex": "W7222996122",
  "pdf_url": "https://arxiv.org/pdf/2610.24580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bnxhtcwyr7mc7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Complex Representatives, Similarity Invariants, and Iterates in $\\bf{U(1,1;\\mathbb{H})}$",
  "title_text": "Complex Representatives, Similarity Invariants, and Iterates in bfU(1,1;ℍ)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/bnxhtcwyr7mc7",
  "venue": "arXiv",
  "versions": []
}