Euclidean extensions of dynamical systems

We consider special Euclidean SE n group extensions of dynamical sys tems and obtain results on the unboundedness and growth rates of trajectories for smooth extensions The results depend on n and the base dynamics consid ered For discrete dynamics on the base with a dense set of periodic points we prove unboundedness of trajectories for generic extensions provided n or n is odd If in addition the base dynamics is Anosov then generically trajec tories are unbounded for all n exhibit square root growth and converge in distribution to a nondegenerate standard n dimensional normal distribution For su ciently smooth SE extensions of quasiperiodic ows we prove that trajectories of the group extension are typically bounded in a probabilistic sense but there is a dense set of base rotations for which extensions are typically unbounded in a topological sense The results on unboundedness are generalised to SE n n odd and to extensions of quasiperiodic maps We obtain these results by exploiting the fact that SE n has the semi direct product structure Gn R where G is a compact connected Lie group and R n is a normal abelian subgroup of This means that our results also apply to extensions by this wider class of groups To appear in Nonlinearity