# Multifractal measures and stability islands in the anisotropic Kepler problem

## Abstract

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.