Euclidean extensions of dynamical systems

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Abstract
We consider special Euclidean ( SE(n)) group extensions of dynamical systems and obtain results on the unboundedness and growth rates of trajectories for smooth extensions. The results depend on n and the base dynamics considered. For discrete dynamics on the base with a dense set of periodic points, we prove the unboundedness of trajectories for generic extensions provided n = 2 or n is odd. If in addition the base dynamics is Anosov, then generically trajectories are unbounded for all n, exhibit square root growth and converge in distribution to a non-degenerate standard n-dimensional normal distribution. For sufficiently smooth SE(2)-extensions of quasiperiodic flows, 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 generalized 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 Γ = G n, where G is a compact connected Lie group and n is a normal Abelian subgroup of Γ. This means that our results also apply to extensions by this wider class of groups.