DISTRIBUTION OF CYCLES FOR ONE-DIMENSIONAL RANDOM DYNAMICAL SYSTEMS

The paper proves (a) and (b) via level‑2 large deviations for the skew product R, plus a samplewise conversion using sub‑exponential bounds on weighted fixed‑point counts; this yields ν̃^ω_n ⇒ δ_{λ_p} and ν^ω_n ⇒ λ_p for m_p‑a.e. ω under (A1)–(A3). In contrast, the model’s solution incorrectly identifies Fix(T^n_ω) with σ^n‑periodic sequences in a length‑n cylinder and invokes periodic‑orbit equidistribution for Hölder potentials on mixing SFTs. That correspondence is false in general (σ^n‑periodicity corresponds to Fix(R^n), not to Fix(T^n_ω)), and φ need not be Hölder under (A1)–(A3) (neutral fixed points), as the paper explicitly addresses via inducing. Hence the model’s proof breaks at its core mapping and regularity steps, while the paper’s argument is correct and complete.