Isochronous sets of invariant control systems

Let G be a connected Lie group with Lie algebra g and Σ = ( G , D ) a controllable invariant control system. A subset A ⊂ G is said to be isochronous if there exists a uniform time T A > 0 such that any two arbitrary elements in A can be connected by a positive orbit of Σ at exact time T A . In this paper, we search for classes of Lie groups G such that any Σ has the following property: there exists an increasing sequence of open neighborhoods ( V n ) n ≥ 0 of the identity in G such that the group can be decomposed in isochronous rings W n = V n + 1 − V n . We characterize this property in algebraic terms and we show that three classes of Lie groups satisfy this property: completely solvable simply connected Lie groups, semisimple Lie groups and reductive Lie groups.