ON ITERATES OF RATIONAL FUNCTIONS WITH MAXIMAL NUMBER OF CRITICAL VALUES

The paper proves Theorem 1.1 (for simple F of degree m ≥ 4, every indecomposable decomposition of F^∘ℓ is equivalent to the canonical ℓ-fold composition) via a curve-genus/reducibility strategy: (i) show Mon(F)=S_m and indecomposability of F for simple maps, (ii) control when F(x)-G(y)=0 has genus 0 or is reducible, obtaining binomial-coefficient constraints on deg G, and (iii) use a Sylvester–Schur prime-divisor argument to conclude that any left factor of an iterate must be F up to Möbius, then iterate this to get the full result . The candidate solution gives a different, group-theoretic proof: using Mon(F)=S_m, it identifies a canonical minimal block system for Mon(F^∘ℓ) by partitioning each regular fiber into F-fibers and shows that the rightmost indecomposable factor must be F up to Möbius; iterating yields the result. The argument hinges on standard monodromy/block-system facts and appears sound. Both reach the same theorem, but by substantially different methods. The paper’s m ≥ 4 hypothesis is necessary (explicit counterexamples exist for m=2,3) and is acknowledged in both accounts .