PIECEWISE QUASICONFORMAL DYNAMICAL SYSTEMS OF THE UNIT CIRCLE

The paper’s main extension theorem (Theorem 4.1) is stated and proved rigorously: when no hyperbolic→parabolic mismatch occurs, adjacent-image lengths are uniformly comparable (estimate (4.5)), yielding a quasisymmetric boundary map and hence a quasiconformal extension via Beurling–Ahlfors; when hyperbolic→parabolic occurs, the best bound is logarithmic (estimate (4.6)), which—by a sharp David extension criterion (Theorem 2.6)—gives a David extension . The model’s Part (2) deviates at two crucial points: (i) it asserts the BA local ratio ρ(x,y) ≲ 1 + C/(1+|log y|) in the H→P case, contradicting the paper’s proven logarithmic growth in the adjacent-interval ratio (4.6) ; and (ii) it misrelates the dilatation threshold to the Beltrami coefficient, using K > 1/(1−ε) instead of the correct K > (2−ε)/ε (since |µ| = (K−1)/(K+1)). The paper’s proof is coherent and uses its “quasiconformal elevator” (Lemma 4.5) and distortion lemmas to derive (4.5) and (4.6) and then invokes Beurling–Ahlfors and the David extension theorem appropriately, whereas the model’s BA-based tail estimate is built on the incorrect small-distortion bound and threshold, leading to a wrong conclusion for the H→P case .