The study of chaotic behavior in scattering problems has, in recent
years, uncovered many fundamental properties and has greatly enriched our
understanding of the dynamics of these systems [Noid * et al.*, 1986;
Jung, 1986; Eckhardt, 1988; Hénon, 1988; Bleher * et al.*, 1990;
Kovács, 1990; Ding * et al.*, 1990; Smilansky, 1992;
Ott & Tél, 1993; Ott, 1993; Lai * et al.*, 1993].
In scattering problems, one is typically interested in the behavior
of the scattering function which embodies the relationship between
the input variable(s) characterizing the initial condition of the
incident trajectory and the output variable(s) characterizing the
trajectory after it exits the scattering region. In many cases, the
scattering function may have exceedingly complex behavior, where the
function is singular on a fractal set of values of the input variables.
This type of behavior is indicative of the presence of a chaotic set
in the dynamics [Noid * et al.*, 1986; Jung, 1986; Hénon, 1988;
Bleher * et al.*, 1990; Kovács, 1990; Ding * et al.*, 1990;
Troll, 1991], and near the singular set tiny changes in
the input variables can produce drastic changes in the output variables.
Scattering systems which exhibit this property are now commonly
referred to as ``chaotic scattering'' systems.

In a study by Bleher * et al.* [1988], it was found that
if a trajectory of a chaotic scattering system can leave the scattering
region in one of several different ways, then the space of initial
conditions corresponding to the various exit modes may be separated
by a boundary which is fractal. Unlike dissipative dynamical systems,
where the attractors are generally bounded, the ``attractors''
in these Hamiltonian systems
are located at infinity. Instead of representing where the trajectories
eventually end up, these ``attractors'' represent * how* the
trajectories exit the system; specifically, they correspond to which
exit the trajectories took to leave the system. (We assume that almost all
trajectories will eventually escape from the system, and that there exists a
set of trajectories which stay in the scattering region for all time,
but its Lebesgue measure is zero.) We call the collection of initial
conditions which eventually leave the system through a particular exit
the * exit basin* of that exit. The existence of fractal boundaries
separating the various basins has an important effect on
the system: final state sensitivity [Grebogi * et al.*, 1983b;
MacDonald * et al.*, 1985].
That is, in the presence of small uncertainty in the input, the measure
of input variable values for which it is not possible to predict
the ultimate exiting mode is anomalously large (as compared to the
case with nonfractal boundaries).

In this paper, we show that not only are the boundaries separating exit
basins typically fractal, but they may also possess the stronger
property that any point which is on the boundary of one exit basin
is also simultaneously on the boundary of all the other exit basins.
This interesting property is known as the * Wada property*.
Wada boundaries are to be contrasted with more usual boundaries such as
those on a map showing the countries of Europe: there is only a finite
number of boundary points that are on the boundary of more than two
countries. But, of course, points on the boundary of two countries are uncountable
(they lie on one dimensional curves). One of the first examples of a set
that has the Wada property was published by Yoneyama [1917], and he attributed the
example to Mr. Wada. For examples in physical systems
showing the Wada property see Grebogi * et al.* [1987a],
Kennedy & Yorke [1991], Nusse & Yorke [1995], and Nusse * et al.* [1995].
However, the examples in these studies involved attractor basins of
dissipative dynamical systems. We show in this paper that Hamiltonian
dynamical systems can exhibit this Wada property, and that it is
typical for chaotic scattering systems with multiple exit modes to have
this property.

The organization of the paper is as follows. In Sec. 2 we provide some relevant definitions, and we also describe the chaotic billiard system that was used in this study. In Sec. 3 we present our numerical results in graphical format showing the apparent Wada property, and we compute the uncertainty dimension of the basin boundaries demonstrating the final state sensitivity of the system. In Sec. 4 we discuss how Wada basin boundaries are formed. Section 5 discusses the characterization of basin boundary points as either accessible or inaccessible. Finally in Sec. 6, we present our conclusions and discuss the physical relevance of our results.