Multifractal measures and stability islands in the anisotropic Kepler problem

The relation between the binary code for the trajectories in the anisotropic Kepler problem (AKP) and the coordinates in the surface of section is investigated. The binary label 0<η<1 is found to be a strictly increasing function of the starting point 0<X<2 on the heavy axis for time-reversal symmetric trajectories, excepting a single island for mass ratios between 1.5 and 1.8 which was discovered by Broucke. The function η(x) was calculated with a step size Δx=0.0002, and the corresponding binary label down to 2-48. Relatively flat portions can be associated with trapping near the unstable Kepler-type orbit for low mass ratios, and with trapping on either side of the light axis for large mass ratios. The f{hook}(α) curves, fractal dimension of the set with Hölder exponent α, is unusually wide in both of these limits. © 1989.