A FAMILY OF NON-VOLTERRA QUADRATIC OPERATORS CORRESPONDING TO PERMUTATIONS

The paper defines V_α as a convex combination of two permutation-based QSOs and proves: (i) Γ maps to and stays at e_m in one step; (ii) for α∈(0,1) the omega-limit is a singleton b∈X; (iii) for α=0 it is a periodic s-cycle on X; (iv) for α=1 it is a singleton, together with ergodicity for all α. These statements (Theorem 3.5 and Corollary 3.6) match the candidate’s conclusions exactly. The candidate’s proof uses a clean decoupling: a scalar recursion x_m' = x_m^2 + (1 − x_m)^2 and a normalized linear Markov evolution y' = (αI + (1−α)P)y, yielding the same asymptotics and ergodicity via spectral arguments and Cesàro averaging. One minor issue in the paper is Proposition 3.3(ii): as stated it describes Fix(V_α) = X ∪ {e_m} without excluding α=1; for α=1 the full set of fixed points is actually {x ∈ S^{m−1} : x_m = 1/2} ∪ {e_m}, not necessarily restricted to X. This is easily corrected and does not affect Theorem 3.5 or ergodicity. Main results and the model’s solution agree; the proofs are different but compatible (paper: monotone/Lyapunov; model: linear-algebraic decoupling) .