A note on parabolic-like maps

The paper’s note establishes that the C1 smoothness of the dividing arc in the definition of parabolic-like maps can be replaced by the condition that ∂Δ \ {z0} is a quasiarc, and it explains why the straightening theorem for degree 2 still goes through by ensuring the relevant boundary in the covering extension is a quasicircle, via a quasiarc ‘completion’ argument using [L1, Prop. 5.3(2)] (see the abstract, Definition 1.1, and Section 3.2.1 in the note ). The candidate solution reaches the same conclusions: Part (A) argues that quasiarc geometry suffices to define the external class and to carry out QC surgery; Part (B) gives a reduction to the classical C1 framework by taking a parabolic-like restriction and then applies Lomonaco’s straightening into Per1(1). The methodologies differ: the paper keeps the proof in the relaxed setting by completing the quasiarc to a quasicircle for the surgery step, whereas the candidate reduces to the original C1 setting by restriction. No substantive conflict is found; the candidate’s quasisymmetric boundary framework is a slightly weaker regularity statement than the paper’s real-analytic external map on S1, but it does not contradict the paper and suffices for surgery.