Random iteration on hyperbolic Riemann surfaces

The paper’s Theorem 1.5 states exactly the two claims about left iterated function systems in Bloch domains—(i) every limit point is constant and (ii) convergence to a constant is equivalent to convergence of the unique fixed points—and derives them by combining a uniform contraction on Hol(X,Ω) (Proposition 2.3) with a general topological result (Theorem 3.1) for uniformly contracting sequences of maps; see Theorem 1.5 and its proof via Proposition 2.3 and Theorem 3.1 as summarized in Corollary 3.3 and Remark 3.2 . The candidate solution proves the same result directly by exhibiting an explicit uniform Lipschitz constant c = tanh(R(Ω,X)/2) using Schwarz–Pick, monotonicity of the Poincaré metric, and a computation on hyperbolic balls via reduction to Δ, then reproducing the (i) and (ii) conclusions. This matches the paper’s statements and logic but supplies a sharper, explicit constant that the paper does not spell out. No logical gaps were found in either argument. Hence, both are correct and essentially address the same theorem via different proofs.