A well-known conjecture in one-dimensional dynamics states, that a (sufficienly smooth) minimal action of a finitely generated group on the circle, should be ergodic with respect to the Lebesgue measure. A standard strategy of proving ergodicity, that is due to Sullivan, is the expansion procedure, and an obstacle its application is the presence of non-expandable points.

The only examples of actions on the circle that possess non-expandable points, that we know, are (up to some modifications) PSL(2,Z) and the Ghys-Sergiescu smooth action of the Thompson group. Both of them satisfy an interesting proper- ty that every non-expandable point is an isolated fixed point for one of diffeo- morphisms.

Recently, Deroin, Navas and myself proved the ergodicity for such actions under an additional assumption of this property. My talk will be devoted to further study of such groups: a joint result with D. Filimonov, stating, that every such group is a subgroup of a Thompson-like one.