  Hybrid (and impact) systems are dynamical systems experiencing both
continuous and discrete transitions. In this work, we derive necessary and
sufficient conditions for when a given differential form is invariant, with
special attention paid to the case of the existence of invariant volumes.
Particular attention is given to impact systems where the continuous dynamics
are Lagrangian and subject to nonholonomic constraints. A celebrated result for
volume-preserving dynamical systems is Poincar\'e recurrence. In order to be
recurrent, trajectories need to exist for long periods of time, which can be
controlled in continuous-time systems through e.g. compactness. For hybrid
systems, an additional mechanism can occur which breaks long-time existence:
Zeno (infinitely many discrete transitions in a finite amount of time). We
demonstrate that the existence of a smooth invariant volume severely inhibits
Zeno behavior; hybrid systems with the "boundary identity property" along with
an invariant volume-form have almost no Zeno trajectories (although Zeno
trajectories can still exist). This leads to the result that many billiards
(e.g. the classical point, the rolling disk, and the rolling ball) are
recurrent independent on the shape of the compact table-top.
