Given a group it is a natural problem to understand and classify its actions by homeomorphisms on the real line. In particular it is interesting to understand which such actions are structurally stable, i.e. rigid under small deformations. I will address these questions for Thompson's group F, a finitely presented group which admits a natural action on the real line by piecewise affine homeomorphisms. We will see that this group turns out to admit a vast family of other "exotic" actions, and I will present a dynamical classification of its actions which allows to deduce some rigidity results. In particular, I will explain that its "standard" action is structurally stable. A tool in the proof of this result is the study of a certain flow on a compact space constructed by B. Deroin, which encodes all actions of a given group on the real line.
This is part of joint works with J. Brum, J. Carnavale, C. Rivas, M. Triestino.