SYMMETRIC RIGIDITY FOR CIRCLE ENDOMORPHISMS WITH BOUNDED GEOMETRY

The paper proves a symmetric rigidity theorem for circle endomorphisms with bounded geometry and Lebesgue-preserving dynamics: if the conjugacy h is symmetric and fixes 1, then h = id. The proof builds on a sharp supremum argument using the set X of points where the stretch ratio |h(I)|/|I| attains its supremum, plus the bounded-geometry Markov structure and the fact that both f and g preserve Lebesgue measure, to deduce Φ = 1 and then h = id (Theorem 1 and its proof sketch; see Theorem 1 statement and outline in the introduction and Section 3: Lemmas 1–3 and the contradiction argument leading to Φ = 1) . By contrast, the model’s approach hinges on two problematic steps: (i) a nontrivial “flattening across siblings” estimate for R(I) = |h(I)|/|I| derived from symmetry and bounded geometry, which is not established and would require a careful pairing of equal-length adjacent subintervals; and (ii) the claim that symmetric ⇒ quasisymmetric ⇒ absolutely continuous on the circle, which the paper itself explicitly warns is false in general for circle homeomorphisms (symmetric or quasisymmetric maps need not be absolutely continuous) . As a result, the model’s derivation of h′ = 1 a.e. and h = id is not justified.