Creation mechanism of Devil's Staircase surface and unstable and stable periodic orbits in the anisotropic Kepler problem

In the two-dimensional Anisotropic Kepler Problem (AKP), a longstanding question concerns the uniqueness of an unstable periodic orbit (PO) for a given binary code (modulo symmetry equivalence). In this paper, a finite-level (N) surface defined by the binary coding of the orbit is considered over the initial-value domain D ₀. A tiling of D ₀ by base ribbons of the surface steps is shown to be proper, i.e., the surface height increases monotonously when the ribbons are traversed from left to right. The mechanism of creating a level-(N+1) tiling from the level-N tiling is clarified in the course of the proof. There are two possible cases depending on the code and the anisotropy. (A) Every ribbon shrinks to a line as $N\to \infty$. Here, the uniqueness holds. (B) When future (F) and past (P) ribbons become tangential to each other, they escape from the shrinking. Then, the initial values of a stable PO (S) and an unstable PO (U) sharing the same code co-exist inside the overlap of the F and P non-shrinking ribbons. This corresponds to Broucke's PO. When the anisotropy is high, only case A is observed; however, as the anisotropy decreases, a bifurcation of the form $U\left(R\right)\to S\left(R\right)+U^{\prime }\left(NR\right)$ occurs along with the emergence of a non-shrinking ribbon. (Here, R and NR denote self-retracing and non-self-retracing POs, respectively). We conjecture that, from a classification based on topology and symmetry, case B occurs only for odd-rank, Y-symmetric POs. We report two applications. First, the classification is applied successfully to the successive bifurcation of a high-rank PO (n = 15), where the above bifurcation is followed by $S\left(R\right)\to S^{\prime }\left(R\right)+S^{\prime \prime }\left(NR\right)$. Second, enhancing the sensitivity to the co-existence of S and U POs through ribbon tiling, we examine the high-anisotropy region. A new symmetry-type POs (O-type) are found and, at γ = 0.2, all POs are shown to be unstable and unique. An investigation of 13648 POs at rank 10 verifies that Gutzwiller's action formula works with amazing accuracy. (paper)