  In this paper, we study pointwise decay estimates in time for Vlasov fields
on non-trapping asymptotically hyperbolic manifolds. We prove optimal decay
estimates in time for the spatial density induced by Vlasov fields on these
geometric backgrounds in dimension two. First, we show exponential decay for
Vlasov fields on hyperbolic space supported away from the zero velocity set. In
contrast, we obtain inverse polynomial decay for general Vlasov fields on
hyperbolic space. In the second part of the article, we prove exponential decay
for Vlasov fields on non-trapping asymptotically hyperbolic manifolds supported
away from the zero velocity set. The proofs are obtained through a commuting
vector field approach. We exploit the hyperbolicity of the geodesic flow in
these geometric backgrounds, by making use of a commuting vector field in the
unstable invariant distribution of phase space.
