Piecewise linear circle maps and conjugation to rigid rational rotations

The paper’s Theorem 2.2 states and proves that a PWL orientation-preserving circle homeomorphism with rational rotation number p/q is conjugate by a PWL map to the rigid rotation r_{p/q} if and only if every break point is periodic; the proof shows that periodic break points force f^q ≡ Id on the circle, and then a PWL conjugacy is constructed (via Lemma 3.1) . The candidate solution proves the same equivalence, using the same core idea—on each component between break-orbit points, f^q is affine and fixes endpoints, hence is the identity there, giving f^q ≡ Id—then builds a conjugacy by solving a piecewise-linear cohomological equation. One minor flaw is the model’s internal justification that every periodic point must have minimal period exactly q; the provided argument is incorrect, though the fact itself is standard for circle homeomorphisms and is explicitly invoked in the paper’s proof (“they must all have the same period, q”) . With that correction, the two proofs are essentially the same in structure, differing mainly in how the conjugacy is constructed after f^q ≡ Id.