Periodic points on the regular and double n-gon surfaces

The paper’s main theorem states that for the regular n-gon and double n-gon translation surfaces, when n ≥ 5 and n ≠ 6, the periodic points are exactly the Weierstrass points that are not singularities of the flat metric, and it outlines the transfer-principle + rational-height proof strategy (Theorem 1.3 and surrounding discussion) . It also notes that these surfaces are hyperelliptic with hyperelliptic involution of derivative −Id, and that nonzero Weierstrass points are periodic because the involution is central in the affine group (Remark 2.9) . The candidate solution matches these points. However, the model incorrectly asserts that for n = 6 (a torus) there are no periodic points; the paper explicitly states that the torus cases (n = 3, 4, 6) have infinitely many periodic points coming from torsion points . Hence, aside from the torus exception, the model’s reasoning aligns with the paper.