Slowly recurrent Collet–Eckmann maps with non-empty Fatou set

The paper proves the stated density result (Theorem 1.3) for slowly recurrent Collet–Eckmann (CE) rational maps via a Benedicks–Carleson–type parameter exclusion in one-dimensional analytic families, coupled with transversality along almost every parameter direction and a Fubini argument to lift to Λ_{d,p′} . The core ingredients include: phase–parameter transversality for 1D families (Lemma 3.1) , outside/free expansion and returns near critical neighborhoods (Lemmas 3.7, 3.9, 3.10) , a strong distortion/large-scale dichotomy (Lemma 3.15) , an escape-position measure estimate (5.1) and exponential tail for escape times (4.2) , and the final measure intersection over critical points plus Fubini to conclude Lebesgue density . By contrast, the model’s solution asserts a multidimensional argument in a local chart of Λ_{d,p′} using ordered critical values and claims uniformly well-conditioned Jacobians for the maps a ↦ f_a^n(v_j(a)) (including control of cross-partials and bounded distortion on parameter balls). This strong, multi-parameter transversality and global bounded-distortion control is not established in the paper (which only proves 1D transversality for non-degenerate directions) . The model also replaces the paper’s uniform Fatou escape (5.1) by an argument via holomorphic motion of attracting basins, which is unnecessary and not justified in the generality used. Consequently, the model’s proof is incomplete on the decisive multi-parameter transversality/distortion step, whereas the paper’s proof is correct and complete.