Notes on the U-Space Iteration for Negative Parameters

This paper investigates the U-space iteration u_{n+1} = u_n^{1/s} e^{-\pi u_n / s^2} for negative parameters s < 0. Let t = -s. The main results are:

1. The local stability boundary t_c ≈ 2.07060658 is derived from the condition |λ| = 1, leading to the transcendental equation t(t-1) = π e^{-(t-1)/(t+1)}.

2. For the initial value u_0 = 0.5, the global convergence boundary is t_c(0.5) ≈ 2.074, which exceeds t_c. The basin boundary curve t_c(u_0) exhibits a U-shaped profile.

3. Near t → t_c^+, the number of iterations diverges as n ∼ 1/|t - t_c|, with the coefficient analytically determined.

4. Complex initial values also converge to the real fixed point, at a slower rate.

These results distinguish local attraction from global convergence in the negative parameter regime.