Euclidean extensions of dynamical systems

Citation
M. Nicol et al., Euclidean extensions of dynamical systems, NONLINEARIT, 14(2), 2001, pp. 275-300
Citations number
14
Categorie Soggetti
Mathematics
Journal title
NONLINEARITY
ISSN journal
09517715 → ACNP
Volume
14
Issue
2
Year of publication
2001
Pages
275 - 300
Database
ISI
SICI code
0951-7715(200103)14:2<275:EEODS>2.0.ZU;2-6
Abstract
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.