The problem presented in this paper is a model related to the collinear three-body problem which is itself a special case of the full three-body problem where three point masses interact under inverse square gravitation. The three-body problem has been known since Newton to be notoriously difficult. The search for a full solution has motivated a great deal of work but was unsuccessful due in part to the possibility of chaotic behavior discovered by Poincare in the 1880's and 1890 's [3].
The cause of complicated behavior in the three-body problem lay in the fact that solutions can pass near triple collision. In 1974 R. McGehee introduced a transformation which allowed him to extend the phase space to triple collision in the collinear three-body problem. McGehee studied the flow on the collision manifold in some detail in order to understand orbits which pass near to triple collision [1]. His analysis is essentially a tool for understanding systems containing triple collisions.
In 1994, Meyer and Wang showed that the structure of the stable manifold for triple-collision in the phase space of the collinear three-body problem is complicated [2]. They also show the existence of symbolic systems which describe the dynamics but to state the symbolic dynamics requires tracking the stable manifold for triple-collision between binary collisions, a notorious problem at best.
In an effort to study the global effect on the dynamics of the collinear three-body problem due to the existence of orbits which pass near triplecollision we present a special model which is a caricature of the collinear three-body problem. The goal of this paper is to describe the dynamics of this model. The main theorem, that the set of sequences of allowed bounces can be described by a sub-shift of finite type, suggests that a similar result might be possible in the collinear three-body problem.
Rather than three point masses on a line we introduce a fixed bumper between two point masses restricted to a line. This bumper has infinite article no.