Global Convergence of a Coupled Nonlinear Iteration System: A Multivariate Generalization

This paper investigates the global convergence of a coupled nonlinear iteration system. Let x_n, y_n in C \ {0} satisfy: x_{n+1} = sqrt(x_n) * e^{-π y_n / 4}, y_{n+1} = sqrt(y_n) * e^{-π x_n / 4} where the principal branch of the square root is taken. Building upon the single-variable convergence theorem established in [1], we prove that for almost all initial points (x_0, y_0) in (C \ {0})^2, lim_{n→∞} x_n = lim_{n→∞} y_n = f where f = e^{-π f / 2} ≈ 0.47454. Furthermore, utilizing the Z-U-V triple duality relations [2], we derive the corresponding convergence in Z-space and V-space.