We consider special Euclidean (SE(n)) group extensions of dynamical systems
and obtain results on the unboundedness and growth rates of trajectories f
or smooth extensions. The results depend on n and the base dynamics conside
red.
For discrete dynamics on the base with a dense set of periodic points, we p
rove 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 conve
rge in distribution to a non-degenerate standard n-dimensional normal distr
ibution.
For sufficiently smooth SE(2)-extensions of quasiperiodic flows, we prove t
hat trajectories of the group extension are typically bounded in a probabil
istic sense, but there is a dense set of base rotations for which extension
s are typically unbounded in a topological sense. The results on unboundedn
ess are generalized to SE(n) (n odd) and to extensions of quasiperiodic map
s.
We obtain these results by exploiting the fact that SE(n) has the semi-dire
ct product structure Gamma = G x R-n, where G is a compact connected Lie gr
oup and R-n is a normal Abelian subgroup of Gamma. This means that our resu
lts also apply to extensions by this wider class of groups.