Kick stability in groups and dynamical systems

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We consider a general construction of 'kicked systems' which extend the framework of classical dynamics. Let G be a group of measure-preserving transformations of a probability space. Given a 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 a given period, and 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. (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 P S L (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. (c) 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 super-recurrent. We use tools of geometric group theory (quasi-morphisms) and symplectic topology (Hofer's geometry).

Original languageEnglish
Pages (from-to)1331-1363
Number of pages33
Issue number5
StatePublished - Sep 2001


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