Nekhoroshev theory and discrete averaging

The paper proves Nekhoroshev stability for analytic, exact symplectic near‑integrable maps under strong convexity, with optimal exponents α=β=1/(2(d+1)) (Theorem 2.1) and an explicit, constructive approach via discrete averaging and an exponentially accurate embedding of near‑identity maps into autonomous Hamiltonian flows (Theorem 5.1) . The candidate solution derives the same exponents by embedding the map as a Poincaré/stroboscopic map of an analytic (d+1)‑degree‑of‑freedom Hamiltonian flow and applying the quasi‑convex Nekhoroshev theorem for flows, then sampling at integer times. This classical route is also acknowledged in the paper as the standard reduction for maps . The only minor gap in the candidate solution is a missing quantitative bound when asserting quasi‑convexity of H0(I,J)=h0(I)+J; a uniform upper bound on |ω| over the domain yields an explicit quasi‑convex constant. Aside from this routine fix, both arguments are sound and reach the same conclusion; the proofs are different in method (flow reduction vs direct discrete averaging).