AI 中文总结
该研究解决三维相关映射的不动点分类与稳定性问题，刻画七个不动点的结构，证明模式点局部渐近稳定、混合与等相关点不稳定，为全测度收敛猜想提供可复现证据。
AI 中文摘要
半个多世纪以来，迭代 Pearson 相关已被用于聚类、序列化和信息可视化，尽管缺乏关于其极限动力学的一般分析理论。我们完全解决了三维相关映射的不动点分类和相对 Lyapunov 稳定性问题。精确的逐行 Gram 分解给出恒等式 $\text{rank}C(A)=\text{rank}(AH_n)$，迫使每个不动点都是奇异的且位于椭圆体的相对边界上。在任意维度中，我们刻画了所有非退化符号值不动点并证明其数量为 $2^{n-1}-1$。对于 $n=3$，我们推导了显式坐标表示并证明自然非退化域恰好包含七个不动点：三个秩 1 模式点、三个秩 2 混合点和一个秩 2 等相关点，在 $S_3$ 的置换作用下形成三个轨道。精确符号 Jacobian 计算证明模式点是局部渐近稳定的，而可允许不变边界曲线实现了扩张乘子，证明混合点和等相关点相对于自然非退化域是 Lyapunov 不稳定的。置换等变性将三个模式 basin 相互映射，并在可测时意味着相等的 Lebesgue 测度。可复现的 basin 计算将每个采样的可允许轨迹分类为收敛到三个模式吸引子之一，频率近乎相等，且在更严格验证下无轨迹级分类不匹配。这些结果建立了三维的完整不动点和相对 Lyapunov 稳定性理论，并为三个模式 basin 的并集具有全测度收敛的猜想提供了强有力的可复现证据。
英文摘要
For more than half a century, iterated Pearson correlation has underpinned methods for network blockmodeling, clustering, seriation, and information visualization. Despite its continued use and the longstanding assumption of convergence, the global convergence problem remained open even in dimension three. We resolve this problem completely for the $3\times3$ correlation map: every admissible orbit is well defined for all forward iterates and converges to one of exactly seven fixed points. In arbitrary dimension, we establish an exact row-wise Gram factorization and the identity $\operatorname{rank}C(A)=\operatorname{rank}(AH_n)$, which gives the precise rank-reduction mechanism and forces every fixed point to be singular. We identify the all-ones matrix as the unique Pearson-degenerate correlation matrix, prove forward invariance of the Pearson-nondegenerate elliptope, and classify the $2^{n-1}-1$ nondegenerate sign-valued fixed points. For $n=3$, we prove a complete analytical fixed-point classification: three rank-one patterned points, three rank-two mixed points, and one rank-two equicorrelation point. We also prove the complete relative Lyapunov stability classification: precisely the patterned points are locally asymptotically stable, while the mixed and equicorrelation points are unstable. The global proof reduces the rank-two dynamics to a one-dimensional projective kernel coordinate; an exact order identity produces monotone projective ratios, excludes nontrivial periodic and recurrent limit sets, and forces convergence to a fixed point. Finally, the initial conditions converging to the four unstable fixed points form a Lebesgue-null set. Hence almost every admissible initial condition converges to a patterned fixed point; the three patterned basins are relatively open and permutation-equivalent, and each has Lebesgue measure exactly one third of that of the elliptope.
Comments45 pages, 3 figures, 2 tables
