Structure of Groups of Circle Diffeomorphisms with the Property of Fixing Nonexpandable Points

We study the structure of groups of circle diffeomorphisms with the property of fixing nonexpandable points. This property generalizes the local expansivity property, and at present there are no known examples of minimal $C^2$ actions of finitely generated groups of circle diffeomorphisms for which this generalized property does not hold.
It turns out that if this property holds for a group action and there is at least one nonexpandable point, then the action admits a rather restrictive characterization. In particular, for such an action, we prove the existence of a Markov partition, and the structure of the action turns out to be similar to that of the Thompson group.