A resolution of non-uniqueness puzzle of periodic orbits in the 2-dim anisotropic Kepler problem: bifurcation U → S + U′

Using binary coding of orbit we introduce a finite level (N) surface over the initial value domain D of 2-dim AKP. It gives a tiling of D by base ribbons. The scheme of the one-time map is studied and the properness of the tiling is proved. This analysis in turn resolves the long standing puzzle in AKP—the non-uniqueness issue of a PO for a given code. We argue that the unique existence of a periodic orbit (PO) for a given binary code generally holds (for inverse anisotropy parameter γ < 8/9) but there is a remarkable exception in which a ribbon with a certain code escapes from shrinking at large N and embodies the Broucke-type stable PO (S). It comes along the bifurcation of an unstable PO (U): U(R) → S(R) + U′(NR) (R for retracing and NR for non-retracing). An analysis based on orbit topology clarifies the pattern of the bifurcation; we give a conjecture that it occurs among odd rank Y-symmetric POs.