So long as it is symmetric, we can easily think of the system as an iterated function map, a common method of representing a chaotic, dynamical system. 
Figure 7 shows one possible representation of the variables, with the first variable, 
, representing the angle around the disc at rebound and the second, , representing the impact/rebound angle relative to the disc.
A subset of these two variables, called the invariant set will map onto themselves.
This set, four members of which are shown in Figures 8 and 9, will be fractal, totally non-attracting
and of measure zero.  This is an interesting inversion of the more normally discussed chaotic systems in which the fractal invariant set is attracting and in fact comprises the basin[s] of attraction.  Note that the totally non-attracting nature of the invariant set is another property of a hyperbolic chaotic scatterer.
Each member of the invariant set can be modelled using symbolic dynamics: the trajectory is labelled based on each of the discs off of which it rebounds.
The set of all such sequences form an uncountable set.
For the four members shown in Figures 8 and 9, the symbolic dynamics will be as follows:
Members of the stable manifold may be likewise represented, except each sequence will have a starting point.  When you consider that a member of the invariant set must "fit" in the boundaries between two basins of attraction, it is apparent that, if perturbed, the trajectory may exit anywhere along the sequence.  Thus it should also be apparent that an infinite number of alternating basins of all three "colours" will exist between any given boundary.
Because of their unstable nature, it is difficult to access members of the invariant set or the stable manifold directly.  The uncertainty exponent is ideally tailored to measure the fractal dimension of this type of system.  Once again using the single impact parameter, b, we perform multiple trials with random impact parameters, perturbing them by a minute amount, , and counting how frequently the number of rebounds off the discs changes, that is, the uncertainty fraction.
Note that even though the system is two dimensional, a single impact parameter is sufficient to measure the fractal dimension of the stable manifold.  This is demonstrated in Figure 10, which shows the basins of attraction plotted as a function of a dual impact parameter,  and .  The stable manifold, which can be seen in the boundaries between the basins, is fractal along only one dimension.
Figure 11 plots the uncertainty fraction, f, as a function of the uncertainty,  for a simulated Gaspard–Rice system.  The slope of the fitted curve returns the uncertainty exponent, , thus the box-counting dimension of the stable manifold is, .  The invariant set is the intersection of the stable and unstable manifolds.
Since the system is the same whether run forwards or backwards, the unstable manifold is simply the mirror image of the stable manifold and their fractal dimensions will be equal.
On this basis we can calculate the fractal dimension of the invariant set:
where D_s and D_u are the fractal dimensions of the stable and unstable manifolds, respectively and N=2 is the dimensionality of the system.  The fractal dimension of the invariant set is D=1.24.