KAM AND GEODESIC DYNAMICS OF BLACKHOLES

The paper’s Theorem 4.4 states that for small stationary perturbations of Kerr, after reparametrizing the frequency ratio by rc and excluding a small neighborhood U of its unique discontinuity, one obtains a closed set C ⊂ R \ U with |R \ (U ∪ C)| = O(ε^{1/2}) on which photon-sphere tori persist with rotation numbers ν ∘ g(Q(rc), Lz(rc)), and that the complementary intervals are Birkhoff regions of instability governed by Aubry–Mather theory . The paper explains the reduction to a 2-DOF Hamiltonian on the photon shell NHIM and the use of the Poincaré return map on {α_θ=0} to obtain an integrable twist map with rotation function ν, then applies KAM and Aubry–Mather machinery, including the O(√ε) measure estimate for twist maps . The candidate solution follows the same plan: persistence of the normally hyperbolic photon shell, construction of the exact symplectic Poincaré map, twist nondegeneracy away from U, Moser–KAM with O(ε^{1/2}) gaps, and Aubry–Mather dynamics in the gaps, yielding the same quantifiers and rotation-number identification. Minor differences are that the candidate imposes axisymmetry (the paper’s statement needs only stationarity) and explicitly invokes exactness of the Poincaré map and a uniform twist bound; these are stronger-than-necessary assumptions but do not contradict the paper’s claims. Overall, the arguments agree in substance and conclusion .