  In this paper we establish a kind of bijection between the orbits of a
polygonal outer billiards system and the orbits of a related (and simpler to
analyze) system called the pinwheel map. One consequence of the result is that
the outer billiards system has unbounded orbits if and only if the pinwheel map
has unbounded orbits. As the pinwheel map is much easier to analyze directly,
we think that this bijection will be helpful in attacking some of the main
questions about polyonal outer billiards.
