Mean index for non-periodic orbits in Hamiltonian systems

The paper proves that for a quasi-periodic orbit the positive/negative upper and lower mean indices all coincide and the limit lim_{T→∞} i1(γ,[0,T])/T exists (Theorem 1.3), by reducing to a quasi-periodic skew-product on the torus and applying Walters' uniform ergodic theorem to a continuous “mean index per block” observable; see the statement of Theorem 1.3 and its reduction to Theorem 3.2 via unique ergodicity of the Kronecker flow and uniform ergodic averages (Theorem 3.1) . The argument controls concatenation errors using the Maslov-type index comparison |ι(M1,γ) − ι(M2,γ)| ≤ 2n (their (5.1)) and bounds the index on bounded intervals (Lemma 2.4), which together handle block gluing and the passage from integer to real times . Negative-time equality is obtained by an explicit identity for the reversed-block observable and invariance of Lebesgue measure under both ±ω rotations . By contrast, the model’s proof uses a subadditive/Kingman approach on the raw Maslov index i1 to get an almost-everywhere limit, but the key step that “upgrades” to a uniform limit for all phases via sup/inf bounds is not justified: from L̃_n/n ≤ ∫b_n/n ≤ Ũ_n/n one only gets l∞ ≤ β ≤ u∞, not l∞ = u∞, without an additional uniform subadditive theorem or continuity/small-defect hypotheses ensuring U_n/n − L_n/n → 0. This gap leaves the model’s conclusion (existence of the limit for every phase and equality of the positive/negative mean index sets) unproven, whereas the paper’s proof supplies exactly the missing uniformity via a continuous observable and Walters’ theorem.