Complex Representatives, Similarity Invariants, and Iterates in $\bf{U(1,1;\mathbb{H})}$

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 $Δ(T^n)=0$.