  We study the dynamics of inertial particles in three dimensional
incompressible maps, as representations of volume preserving flows. The
impurity dynamics has been modeled, in the Lagrangian framework, by a
six-dimensional dissipative bailout embedding map. The fluid-parcel dynamics of
the base map is embedded in the particle dynamics governed by the map. The base
map considered for the present study is the Arnold-Beltrami-Childress (ABC)
map. We consider the behavior of the system both in the aerosol regime, where
the density of the particle is larger than that of the base flow, as well as
the bubble regime, where the particle density is less than that of the base
flow. The phase spaces in both the regimes show rich and complex dynamics with
three type of dynamical behaviors - chaotic structures, regular orbits and
hyperchaotic regions. In the one-action case, the aerosol regime is found to
have periodic attractors for certain values of the dissipation and inertia
parameters. For the aerosol regime of the two-action ABC map, an attractor
merging and widening crises is identified using the bifurcation diagram and the
spectrum of Lyapunov exponents. After the crisis an attractor with two parts is
seen, and trajectories hop between these parts with period 2. The bubble regime
of the embedded map shows strong hyperchaotic regions as well as crisis induced
intermittency with characteristic times between bursts that scale as a power
law behavior as a function of the dissipation parameter. Furthermore, we
observe riddled basin of attraction and unstable dimension variability in the
phase space in the bubble regime. The bubble regime in one-action shows similar
behavior. This study of a simple model of impurity dynamics may shed light upon
the transport properties of passive scalars in three dimensional flows. We also
compare our results with those seen earlier in two dimensional flows.
