  We call internal-wave billiard the dynamical system of a point particle that
moves freely inside a planar domain (the table) and is reflected by its
boundary according to this rule: reflections are standard Fresnel reflections
but with the pretense that the boundary at any collision point is either
horizontal or vertical (relative to a predetermined direction representing
gravity). These systems are point particle approximations for the motion of
internal gravity waves in closed containers, hence the name. For a class of
tables similar to rectangular trapezoids, but with the slanted leg replaced by
a general curve with downward concavity, we prove that the dynamics has only
three asymptotic regimes: (1) there exist a global attractor and a global
repellor, which are periodic and might coincide; (2) there exists a beam of
periodic trajectories, whose boundary (if any) comprises an attractor and a
repellor for all the other trajectories; (3) all trajectories are dense (that
is, the system is minimal). Furthermore, in the prominent case where the table
is an actual trapezoid, we study the sets in parameter space relative to the
three regimes. We prove in particular that the set for (1) has positive measure
(giving a rigorous proof of the existence of Arnol'd tongues for internal-wave
billiards), whereas the sets for (2) and (3) are non-empty but have measure
zero.
