CONSTRUCTION OF MINIMAL COCYCLES ARISING FROM SPECIFIC DIFFERENTIAL-EQUATIONS

Blending methods of Topological Dynamics and Control Theory, we develo p a new technique to construct compact-Lie-group-valued minimal cocycl es arising as fundamental matrix solutions of linear differential equa tions with recurrent coefficients subject to a given constraint. The p recise requirement on the coefficients is that they belong to a specif ied closed convex subset S of the Lie algebra L of the Lie group. Our result is proved for a very thin class of cocycles, since the dimensio n of S is allowed to be much smaller than that of L, and the only assu mption on S is that L-0(S) = L, where L-0(S) is the ideal of L(S) gene rated by the difference set S - S, and L(S) is the Lie subalgebra of L generated by S. This covers a number of differential equations arisin g in Mathematical Physics, and applies in particular to the widely stu died example of the Rabi oscillator.