Nonlinearity. 14 (2001) 275-300
Matthew Nicol, Ian Melbourne and Peter Ashwin
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 on the underlying dynamics on the base.
For discrete dynamics on the base with a dense set of periodic points, we prove 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 nondegenerate 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 hold also for SE(n) (n odd) and for extensions of quasiperiodic maps.
We obtain these results by exploiting the fact that SE(n) has the
semi-direct product structure $\Gamma=G+Rn$ where G is a
compact
connected Lie group and Rn is a normal abelian subgroup of $\Gamma$.
This means that our results also apply to extensions by this wider class of
groups.