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PreprintCC BY 4.0arXiv

Complex Representatives, Similarity Invariants, and Iterates in bfU(1,1;ℍ)

  • Bingzhe Hou
  • Cailing Yao
  • Jiaxing He
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Abstract

We study the similarity classes of matrices in U(1,1;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;ℍ) is similar, via a matrix in U(1,1;ℍ), to a complex matrix. Explicit complex representatives and matrices realizing these similarities are constructed for the different types. We then introduce the invariants 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;ℍ). Finally, using the complex representation of quaternionic matrices and Chebyshev polynomials, we derive a fourth-order recurrence for a real sequence (qn) satisfying Δ(Tn)=qn2Δ(T). Together with recurrence relations for the relevant matrix entries, this identity yields a complete classification of the quaternionic Möbius transformation associated with Tn, including the degenerate cases in which Δ(Tn)=0.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026complex,
  title = {{Complex Representatives, Similarity Invariants, and Iterates in $\bfU(1,1;\mathbbH)$}},
  author = {Bingzhe Hou and Cailing Yao and Jiaxing He},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24580},
  eprint = {2610.24580},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24580},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Complex Representatives, Similarity Invariants, and Iterates in bfU(1,1;ℍ)
AU  - Bingzhe Hou
AU  - Cailing Yao
AU  - Jiaxing He
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24580
UR  - https://arxiv.org/abs/2610.24580
AB  - We study the similarity classes of matrices in U(1,1;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;ℍ) is similar, via a matrix in U(1,1;ℍ), to a complex matrix. Explicit complex representatives and matrices realizing these similarities are constructed for the different types. We then introduce the invariants 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;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.
ER  -