REFINEMENTS OF FRANKS’ THEOREM AND APPLICATIONS IN REEB DYNAMICS

The paper proves Theorem 1 and its reversible refinement (Theorem 2) as stated, using rotation-number forcing (Franks/Le Calvez) and a maximal-isotopy/foliation argument to produce distinct rotation numbers when needed, then selecting denominators coprime to n0. The candidate solution reaches the same conclusions: in the two-rotation-number case it uses the same forcing; in the constant-rotation case it gives a shorter route via Franks’ 1992 theorem on infinitely many interior periodic points. Minor fix: in the reversible gcd-refinement, a q-prime-periodic orbit of g=f^k yields a symmetric periodic orbit of f whose prime period divides kq (not necessarily equals kq). This correction does not affect the conclusion that infinitely many symmetric prime periods remain coprime to n0. Overall, the paper is correct and complete, and the model’s solution is essentially correct after that small repair (Theorem 1 statement and interpretation match the paper’s union-of-Per sets, which avoids claiming infinitely many distinct periods) .