THE MULTIPLE ERGODICITY OF NONDISCRETE SUBGROUPS OF Diffω (S1)

We deal with nondiscrete subgroups of Diff(omega) (S-1), the group of orientation-preserving analytic diffeomorphisms of the circle. If Gamma is such a group, we consider its natural diagonal action (Gamma) over tilde on the ndimensional torus T-n. A complete characterization of those groups Gamma whose corresponding (Gamma) over tilde -action on T-n is not piecewise ergodic (see Introduction) for all n is an element of N is obtained (see Theorem A). Theorem A can also be interpreted as an extension of Lie's classification of Lie algebras on S-1 to general nondiscrete subgroups of S-1