Kick stability in groups and dynamical systems

We consider a general construction of 'kicked systems' which extend the fra mework of classical dynamics. Let G be a group of measure-preserving transf ormations of a probability space. Given a one-parameter/cyclic subgroup (th e flow), and any sequence of elements (the kicks) we define the kicked dyna mics on the space by alternately flowing with a given period, and then appl ying a kick. Our main finding is the following stability phenomenon: the ki cked system often inherits recurrence properties of the original flow. We p resent three main examples. (a) 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. (b) G is a discrete subgroup of PSL (2, R) acting on the unit tangent bundl e 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 i n G. (c) G is the group of Hamiltonian diffeomorphisms of a closed symplectic ma nifold. 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 a nd is thus super-recurrent. We use tools of geometric group theory (quasi-morphisms) and symplectic top ology (Hofer's geometry).