The multiple ergodicity of nondiscrete subgroups of ${\rm Diff}^\omega(S^1)$

We deal with nondiscrete subgroups of ${\rm Diff}^\omega(S^1)$, the group of orientation-preserving analytic diffeomorphisms of the circle. If $\Gamma$ is such a group, we consider its natural diagonal action $\widetilde\Gamma$ on the $n$-dimensional torus $\mathbb T^n$. A complete characterization of those groups $\Gamma$ whose corresponding $\widetilde\Gamma$-action on $\mathbb T^n$ is not piecewise ergodic (see Introduction) for all $n\in\mathbb N$ 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 $S^1$.