  Hybrid dynamical systems are systems which undergo both continuous and
discrete transitions. As typical in dynamical analysis, an essential goal is to
study the long-term behavior of these systems. In this work, we present two
different novel approaches for studying these systems. The first approach is
based on constructing an analog of the Frobenius-Perron (transport) operator
for hybrid systems. Rather than tracking the evolution of a single trajectory,
this operator encodes the asymptotic nature of an ensemble of trajectories. The
second approach presented applies to an important subclass of hybrid systems,
mechanical impact systems. We develop an analog of Lie-Poisson(-Suslov)
reduction for left-invariant impact systems on Lie groups. In addition to the
Hamiltonian (and constraints) being left-invariant, the impact surface must
also be a right coset of a normal subgroup. This procedure allows a reduction
from a $2n$-dimensional system to an $(n+1)$-dimensional one. We conclude the
paper by presenting numerical results on a diverse array of applications.
