Fractal dimension of higher-dimensional chaotic repellors

Using examples we test formulae previously conjectured to give the fractal information dimension of chaotic repellors and their stable and unstable manifolds in “typical” dynamical systems in terms of the Lyapunov exponents and the characteristic escape time from the repellor. Our main example, a three-dimensional chaotic scattering billiard, yields a new structure for its invariant manifolds. This system also provides an example of a system which is not typical and illustrates how perturbation to the system restores typicality and the applicability of the dimension formulae.