Ergodic theorem for the differential equations with interaction

The paper states and (sketch-)proves that for equal-weight discrete initial data µ0 ∈ K̊n and an invariant measure Θn, the average braid invariants satisfy limt→∞ mk(t)/t^k = (1/k!) (E(G|In))^k. Its route is via induction and a continuous Toeplitz-type estimate, combined with an ergodic theorem on (K̊n,Φn,Θn) (see the definition of mk and G, and the ergodic statement (12) in the paper , and Theorem 5.2 ). The candidate solution instead establishes the exact identity mk(t) = (1/k!)[∫0^t G(µs) ds]^k by (i) replacing dφ with its density F and (ii) factorizing the outer µ0-integrals and the time-ordered simplex integral, then applies Birkhoff–Khinchin. This derivation is correct and shorter. The paper’s proof is directionally correct but omits some qualifiers (e.g., “Θn-a.e. µ0”) and technical details (boundedness/L1 of G, diagonal convention for F). Under the paper’s standing invariance assumptions and the usual ergodic-theoretic measurability/integrability conditions, both arguments yield the same limit.