Non-Existence of Wandering Intervals for Asymmetric Unimodal Maps

The paper proves the Main Theorem (no wandering intervals for all f ∈ U2) via a closest-return scheme supported by three precise lemmas: (i) Lemma A: the asymmetry ratio ηk tends to 0 (derived using cross-ratio distortion and a one-sided Koebe principle), (ii) Lemma B: closest pre-returns eventually occur on the side with larger critical order and yield a quantitative lower bound, and (iii) Lemma C: a specific decay estimate ηk ≤ C ηmax(ℓ+,ℓ−) k−1. These are explicitly stated and proved (including Theorems for cross-ratio and one-sided Koebe) and lead to a quantitative contradiction between exponential lower and upper bounds on distances to the critical value, completing the proof . In contrast, the model’s argument hinges on an unsupported Lemma C asserting uniform multiplicative length expansion between successive closest returns (|I_k| ≥ λ|I_{k−1}| with λ>1), which is neither proved nor implied by the cited cross-ratio/Koebe tools and is not used in the paper. The paper’s Lemma C is qualitatively different (it controls ηk, not interval lengths) and the final contradiction in the paper does not rely on any uniform length expansion. Hence the model’s proof is flawed at a critical step, while the paper’s proof is coherent and complete.