Arnold diffusion and geodesic dynamics of blackholes

The paper establishes NHIM persistence near the unstable circular orbit, constructs the scattering map for the perturbed flow, derives a Melnikov generating function and first-order jump formula for p_ϕ, and then applies a standard diffusion/shadowing scheme to obtain an O(1) drift in L_z while keeping L^2 ∈ (L^2_isco, L^2_homo) (Theorems 3.4 and 3.5). The scattering map is shown to be symplectic and O(ε)-close to the identity in C^1, with jump formula 1/ε (p_ϕ(z+) − p_ϕ(z−)) given by the Melnikov integral, under nondegenerate critical points of the Melnikov potential on an open set U . The diffusion theorem for stationary perturbations asserts that, if the RHS of the jump formula is nonzero at some Ξ0 ∈ U, then there is ρ > 0 (independent of ε) and an orbit with |L_z(T) − L_z(0)| > ρ and L^2(t) staying between L^2_isco and L^2_homo over the entire itinerary . The NHIM setup, the use of a compact cut-off to apply Fenichel theory, and the construction of the scattering map are explicitly laid out in the stationary Schwarzschild setting , with the Melnikov/scattering map proof details given in Appendix A.3 . The candidate solution mirrors this scheme: persistence of N_ε and W^{s,u}(N_ε), Melnikov-based scattering generating function, selection of a compact K ⊂ U with |M| ≥ m0, iterative use of S_ε − Id = O(ε) to produce n = O(1/ε) steps, and a standard NHIM shadowing/transition-map lemma to realize the discrete itinerary. The added Poisson-bracket estimate to control L^2 is a routine compactness argument consistent with the paper’s claims. The only stylistic difference is that the paper cites [GLS] for the shadowing mechanism, whereas the candidate cites windows/transition-map results; both yield the same diffusion mechanism on NHIMs. Accordingly, both are correct and substantially the same proof at the level of ideas and structure.