Parabolic Saddles and Newhouse Domains in Celestial Mechanics

The paper proves Theorem 1.1: there is a Newhouse domain N ⊂ (0,1/2] (with 0 in its closure) for the McGehee-conjugated family P_μ, and for every μ ∈ N there is a basic set Λ_μ with a generically unfolding homoclinic tangency; moreover, Λ_μ is homoclinically related to the parabolic fixed point at infinity O. This is stated explicitly and proved via (i) the McGehee normal form P_μ(x,y) = (x + x^3(y + r_μ), y − x^3(x + s_μ)) with DP_μ(O)=Id and analytic W^{s,u}(O), (ii) a sequence μ_n → 0 of quadratic homoclinic tangencies that unfold generically (Proposition 2.5), and (iii) a renormalization near the tangency converging to the conservative (critical) Hénon map, enabling a direct application of Duarte’s theorem to produce Newhouse windows, together with an argument that the resulting basic sets are homoclinically related to O (Section 2.5.2) . The candidate solution reproduces the same architecture: McGehee normal form and invariant manifolds, existence of small-μ quadratic tangencies that unfold generically, application of Duarte’s parametric conservative Newhouse theorem to obtain open windows accumulating at the tangency, and homoclinic relation of the constructed basic sets with O. Minor differences are non-substantive: the model cites broad “global results” (e.g., GMS) and mentions a curve of cubic tangencies, whereas the paper derives transverse intersections and secondary quadratic tangencies using explicit separatrix-splitting asymptotics (Appendix A) and its Shilnikov-map machinery; also, the paper isolates an involutive symmetry hypothesis P_μ^{-1}∘R = R∘P_μ in its general statement, automatically satisfied in the RPC3BP, which the model does not spell out . These differences do not affect the correctness of the main conclusion.