Periodic Points of Rational Functions of Large Degree over Finite Fields

The paper’s Theorem 1.1 states exactly the target claim—if gcd(q,d)=1 and d≥2, then as j→∞ the averages P(q,d,j) and R(q,d,j) of periodic-point proportions over degree-d polynomials and rational maps on P^1(F_{q^j}) both tend to 0—and proves it via (i) an Effective Image Size Theorem, (ii) a specialization/uniformity theorem (Theorem 3.3), and (iii) generic iterated Galois groups with vanishing fixed-point proportions, together with a counting/Lang–Weil excision to control exceptional specializations . The candidate solution reproduces the same architecture: periodic points lie in images of iterates; image sizes are governed by fixed-point proportions of iterated monodromy groups; for generic degree-d maps those groups are iterated wreath products; fixed-point proportions therein decay to 0; Lang–Weil makes the exceptional set negligible. Differences are mostly expositional (e.g., framing the bound as limsup ≤ FPP(W_n) before sending n→∞), but substantively the proofs coincide.