Pairs of Involutions of Glancing Hypersurfaces

(1.1) F̂ : x1 = 0, Ĝ : ξ2 = ξ 2 1 + x1 under a (C) smooth change of coordinates; Melrose’s argument also shows that all real analytic glancing hypersurfaces are equivalent to the above normal form by formal symplectic maps. It was proved by Oshima [6] for n ≥ 3 and by the second author [3] for n ≥ 2 that for some pairs of real analytic glancing hypersurfaces, the normal form cannot be achieved by any convergent symplectic map.