We say a basin boundary point is * accessible* from
a particular basin, if there is another point in the interior
of the particular basin which can be connected to by
a finite length curve that contains no boundary points except
. We say
is * inaccessible from a particular
basin* if this is not possible, and if
is inaccessible from every basin we say that
is * inaccessible*.
(Accessible and inaccessible basin boundary points were first
discussed in Grebogi * et al.* [1986, 1987c].) In this section
we characterize the accessibility and inaccessibility of
basin boundary points for our system. In particular, we show
the following: (i) All basin boundary points that are
accessible from the red region lie on the stable manifold
of the fixed point (1/3,0) (and similarly for the blue region and
the fixed point (0,0), and for the green region and the fixed point
(2/3,0)). (ii) All basin boundary points not on the stable manifold
of one of the three fixed points are inaccessible. (iii) The
stable manifolds of the fixed points (0,0), (1/3,0), (2/3,0) are each dense
in the basin boundary (i.e., every neighborhood of a basin
boundary point contains part of the stable manifolds of all
three fixed points, and the basin boundary is thus the closure
of the stable manifold of any of the fixed points).

To proceed we introduce * trapping regions* for each of the
three exits. Trapping regions are regions in the surface of section where all
trajectories that exit the system will ultimately enter, and once
they entered, the trajectories will not leave the trapping regions
until they exit the scattering region (and hence the system).
In order to find appropriate trapping regions, we look at how the
trajectories leave the exits. We concentrate on one of
the exits, say exit I, and the results will apply to the other two
exits using symmetry arguments. We first note the three
largest (red) connected regions in the basin of exit I (see
Fig. 6); these three regions are shown schematically
in Fig. 8 and are labeled A, B, and C. Any trajectory
that leaves through exit I will bounce off either disk 1, 2, or 3 in
its last collision within the scattering region. The orbit locations in
the surface of section corresponding to the last collision in the
scattering region before going through exit I are
the three shaded areas in regions A, B, and C shown in
Fig. 8. Specifically, the shaded area in region
A is the set of initial conditions that starts from disk 1 and leaves
directly through exit I with no additional collisions within the scattering
region, while the shaded area in region B is the set that starts from disk 2
and immediately leaves through exit I, and the shaded area in region C is
the set that starts from disk 3 and immediately leaves through exit I.
The union of regions A, B, and C form what we called a * trapping set*
for exit I. Every point in the trapping set eventually leaves through
exit I (the trapping set is red in Fig. 6), and, more
importantly, any point that leaves through exit I must eventually
enter the trapping set. The latter follows since if a trajectory
leaves through exit I, it must spend its last iterate in one of
the shaded areas of Fig. 8, and these are located
in the trapping set for exit I. Similar trapping sets
exist for the other two exits (they can be seen in
Fig. 6, where the three largest green swatches correspond
to the trapping set for exit II, and the three largest blue
swatches correspond to the trapping set for exit III.)

We now show that the boundaries of the trapping sets (aside
from the vertical lines s = 0, 1/3, 2/3) lie
on finite segments of the stable manifolds of the fixed points.
While the ``sprinkler algorithm'' used earlier in the section
generated stable and unstable manifolds of the invariant set, it
does not generate stable and unstable manifolds for specific points
within the invariant set. For a picture of the stable
manifold of the fixed points, we employ an algorithm introduced in
You * et al.* [1991] which generates very accurate approximations
of stable and unstable manifolds[2].
Since the system we are studying has three-fold symmetry, we only need to
compute the stable manifold for one of the fixed points. We computed
stretches of the stable manifold corresponding to the fixed point
(1/3,0) as shown in Fig. 9.
From Fig. 9, we see two branches of
the stable manifold of the fixed point (1/3,0),
one coming towards the fixed point from the upper left and one coming
towards the fixed point from the lower right. We observe from
comparison of Figs. 8 and 9
that the upper left and lower right segments are boundaries of the regions A and B
respectively. Portions of the two segments of
the stable manifold immediately to the left and right of the fixed point
(1/3,0) are then iterated backwards to obtain the segments
bounding the top and bottom of region C. For clarity, segments of the
stable manifold are shown either dotted or solid to indicate how they are
iterated forward in time. For example, the top segment bounding region
C (solid) gets mapped to the upper portion of the curve (solid) bounding
region A and next into the middle segment bounding region B (solid).
Similarly, the bottom segment bounding region C (dotted) gets mapped
to the lower portion of the curve (dotted) bounding region B and next
into the middle segment bounding region A (dotted). Thus the boundary of
trapping set I lies on finite segments of the stable manifold of the fixed
point (1/3,0).

In Fig. 10, we have superimposed the segment of
an unstable manifold of the fixed point (1/3,0) computed
using the algorithm of You * et al.* [1991] over the color basin
picture. It is clear that the manifold crosses all three basins.
From this fact, the following argument of Kennedy & Yorke [1991]
shows that points on the stable manifold of (1/3,0) are
Wada and are accessible from the red basin. Denote the three different
basins by _{1}, _{2}and _{3}, and
let B_{1}, B_{2}and B_{3} be subsets corresponding to
the _{i}'s.
Suppose p is a saddle fixed point on the basin boundary, and the unstable
manifold of p crosses portions of all the three different basins.
Examining successive pre-images of the B_{i}'s, we note that they approach the
stable manifold of p because the distance between points on the unstable
manifold contracts under the inverse map. Similarly, pre-images
of the B_{i}'s will expand along
the direction of the stable manifold of p as shown in Fig. 11.
Thus, successive pre-images of the B_{i}'s get arbitrary close to the stable
manifold of p, and every point on the stable manifold of p is on the
boundary of the B_{i}'s for each i [Kennedy & Yorke, 1991].
Thus, every point on the stable manifold of p is a Wada point.
On the other hand, we emphasize that the boundaries of the trapping
region for red are accessible from the red region by our choice of the
trapping region.

Now consider an arbitrary boundary point .
We consider two possibilities: Either (a) the boundary point
iterates to
the boundary of a trapping region in a finite number m of iterates
(i.e., is on the stable manifold of the
appropriate fixed point), or (b) it does not. In case (a)
f^{m}() is an accessible boundary
point. Without loss of generality, suppose f^{m}() is accessible from the red basin, then there is a curve in the red
region connecting it to an arbitrarily nearby interior point of the red region.
Iterating f^{m}(),
and the curve connecting
f^{m}()
and backward m steps, we have
a curve connecting
to the interior point f^{-m}().
Thus is accessible, and any point point
accessible from red is on the stable manifold of (1/3,0).
Also, since the boundary is Wada, any boundary point not accessible
from the red basin has an interior red point arbitrarily nearby. Moving
along a straight line from the interior red point toward the boundary
point that is not accessible from red, we must encounter a first
boundary point, and this is by definition a boundary point
accessible from the red region. This red accessible boundary
point is on the stable manifold of (1/3,0). Hence
any boundary point inaccessible from red is visited arbitrarily
closely by the stable manifold of (1/3,0); i.e., this
stable manifold is dense in the basin boundary.

Now consider case (b). In this case the orbit from the boundary
point remains in the scattering region
for all time but is not on the stable manifold of one of the fixed points. We note
that, as shown in Fig. 12, the invariant set lies within
the shaded regions bounded by fixed point stable and unstable
segments. As shown in Fig. 12, for a sufficient
number of iterates m, the point f^{m}()
will lie in the interior of one of the shaded regions. To see this, recall that
iterates of the boundary points approach the invariant set. Thus
they must either (i) come near the shaded region of Fig. 12
and stay near it without entering, (ii) stay on its boundary
without entering, or (iii) enter it. Possibilities (i) and
(ii) can be ruled out as follows. The orbit cannot fall on
the fixed point stable manifold segments bounding the stable
region since we are considering case (b). Also iterates of
cannot be close to and outside
of the shaded region by being near these stable manifold segments, since they
would then be in a trapping region and
is a boundary point. On the other hand, consideration of the dynamics of the map
shows that iterates of points on or sufficiently near the
unstable manifold boundaries of the shaded region map to
its interior. Thus the orbit from enters the
shaded region of Fig. 12.

Using the same construction and notation as in Sec. 4, we recall
that iterates of the line segment L() contained within
D() expand exponentially with time and
align with the unstable manifold of the invariant set. Also,
for large enough m, f^{m}() will be in
the interior of the shaded region of Fig. 12, and
f^{m}(L()) will be long enough that
the upper right segment of f^{m}(L()) starting
from f^{m}()
and the lower left segment starting from
f^{m}()
*both* cross segments of the fixed point
stable manifolds bounding the shaded region in which
f^{m}()
lies (see Fig. 12). These stable manifold segments
are Wada (they are part of the trapping region boundaries) and hence
possess red, blue, and green striped regions accumulating on
them. Thus iterating backwards to the original
boundary point , we see that there are
red, blue, and green striped regions in D() on *both* sides of
.
Since can be arbitrarily small,
red, blue, and green stripes are arbitrarily close to on both sides of . Hence
is inaccessible, since any curve
from an interior point drawn to
must always cross such stripes (in fact, it must cross an
infinite number of stripes).