An Entire Function Theory for U-Space Dynamics

An entire function theory is established for the U-space iteration. Through the
transformation w = ln u, the U-space iteration is equivalently converted into
the iteration of the entire function f(w) = a w + b e^w, where a = 1/s,
b = -π/s², and s ∈ ℂ\{0,1}. In the w-coordinate, the critical points are
w_c = ln(s/π) + 2πik (k ∈ ℤ), the fixed point is w* = -W₀(π/(s(s-1))),
and the multiplier is λ(s) = [1-(s-1)W₀(π/(s(s-1)))]/s. The existence and
attractivity of the period-2 orbit {±iπ/2} at s = -2 are proved analytically,
and it is shown that this is the unique parameter value admitting a period-2
orbit of this pure imaginary form. For |s| < 1, the existence of a completely
escaping Fatou component with geometric escape rate |1/s| is established. In
the complex parameter plane, the stability boundary |λ(s)| = 1 is traced
numerically from s ≈ -2.0706 on the real axis through the W₀ branch point at
s = 1/2 + i√(πe - 1/4), delineating the attracting and repelling regimes. An
additional escaping mechanism driven by the exponential term e^w is identified
for parameters with |s| > 1, where critical orbits may escape despite |a| < 1.
Since f(w) = a w + b e^w is an entire function of finite order, the Eremenko-
Lyubich theorem excludes wandering domains for all parameters. The U-space and
S-space are connected through the U-S relation, linking the transcendental
dynamics of the U-space to the algebraic dynamics of the S-space.