SYMBOLIC DYNAMICS IN THE ELLIPTIC ISOSCELES RESTRICTED THREE BODY PROBLEM

The paper proves that for sufficiently large angular momentum G, the Poincaré map ψ on the pericenter section Σ+ admits an invariant set S topologically conjugate to the full shift on N^Z (a Smale horseshoe with infinitely many symbols), see Theorem 1.1 and the definition of Σ+ in the text . The proof proceeds by a conformally symplectic scaling that exhibits a parabolic periodic orbit at infinity and treats the system as a fast non-autonomous perturbation of Kepler’s problem . Then Theorem 2.1 establishes an exponentially small but nonzero splitting with an explicit first-order formula (2.8) yielding at least two transverse homoclinic connections to the infinity orbit . Finally, a Moser-type construction produces the symbolic dynamics and the conjugacy to the shift (Theorem 2.4) . The candidate solution follows this blueprint accurately overall. Two notable inaccuracies are: (i) it wrongly infers “infinitely many” primary homoclinic intersections from the Bessel factor, whereas the paper’s first-order distance uses J1(1) as a constant and guarantees at least two intersections via the sine factor in (2.8) ; and (ii) it describes the infinity orbit as “normally hyperbolic,” while it is parabolic in this setting . These do not affect the final conclusion, since countably many symbols arise from the time-of-flight coding near infinity (Theorem 2.4) rather than from having infinitely many distinct primary intersections .