  We study the dynamics of a collisionless kinetic gas whose particles follow
future-directed timelike and spatially bound geodesics in the exterior of a
sub-extremal Kerr black hole spacetime. Based on the use of generalized
action-angle variables, we analyze the large time asymptotic behavior of
macroscopic observables associated with the gas. We show that, as long as the
fundamental frequencies of the system satisfy a suitable non-degeneracy
condition, these macroscopic observables converge in time to the corresponding
observables determined from an averaged distribution function. In particular,
this implies that the final state is characterized by a distribution function
which is invariant with respect to the full symmetry group of the system, that
is, it is stationary, axisymmetric and Poisson-commutes with the integral of
motion associated with the Carter constant. As a corollary of our result, we
demonstrate the validity of the strong Jeans theorem in our setting, stating
that the distribution function belonging to a stationary state must be a
function which is independent of the generalized angle variables. An analogous
theorem in which the assumption of stationarity is replaced with the
requirement of invariance with respect to the Carter flow is also proven.
Finally, we prove that the aforementioned non-degeneracy condition holds. This
is achieved by providing suitable asymptotic expansions for the energy and
Carter constant in terms of action variables for orbits having sufficiently
large radii, and by exploiting the analytic dependency of the fundamental
frequencies on the integrals of motion.
