  We consider a general construction of ``kicked systems''. Let G be a group of
measure preserving transformations of a probability space. Given its
one-parameter/cyclic subgroup (the flow), and any sequence of elements (the
kicks) we define the kicked dynamics on the space by alternately flowing with
given period, then applying a kick. Our main finding is the following stability
phenomenon: the kicked system often inherits recurrence properties of the
original flow. We present three main examples. 1) G is the torus. We show that
for generic linear flows, and any sequence of kicks, the trajectories of the
kicked system are uniformly distributed for almost all periods. 2) G is a
discrete subgroup of PSL(2,R) acting on the unit tangent bundle of a Riemann
surface. The flow is generated by a single element of G, and we take any
bounded sequence of elements of G as our kicks. We prove that the kicked system
is mixing for all sufficiently large periods if and only if the generator is of
infinite order and is not conjugate to its inverse in G. 3) G is the group of
Hamiltonian diffeomorphisms of a closed symplectic manifold. We assume that the
flow is rapidly growing in the sense of Hofer's norm, and the kicks are
bounded. We prove that for a positive proportion of the periods the kicked
system inherits a kind of energy conservation law and is thus superrecurrent.
We use tools of geometric group theory and symplectic topology.
