Kick stability ingroups and dynamical systems

Abstract

We consider a general construction of `kicked systems' which extend theframework of classical dynamics. Let G be a group ofmeasure-preserving transformations of a probability space. Given aone-parameter/cyclic subgroup(the flow), and any sequence of elements (the kicks) we define the kickeddynamics on the space by alternately flowing with a given period, andthen applying a kick. Our main finding is the following stabilityphenomenon: the kicked system often inherits recurrence properties ofthe original flow. We present three main examples.(a) G is the torus. We show that for generic linear flows, and anysequence of kicks, the trajectories of the kicked system are uniformlydistributed for almost all periods.(b) G is a discrete subgroup of PSL(2,ℝ) acting on the unittangent bundle of a Riemann surface. The flow is generated by a singleelement of G, and we take any bounded sequence of elements of G asour kicks. We prove that the kicked system is mixing for all sufficientlylarge periods if and only if the generator is of infinite order and is notconjugate to its inverse in G.(c) G is the group of Hamiltonian diffeomorphisms of a closedsymplectic manifold. We assume that the flow is rapidly growing in thesense of Hofer's norm, and the kicks are bounded. We prove that for apositive proportion of the periods the kicked system inherits a kind ofenergy conservation law and is thus super-recurrent.We use tools of geometric group theory (quasi-morphisms)and symplectic topology (Hofer's geometry).