Abstract: A group G of homeomorphisms of the real line is locally moving if every open interval supports a subgroup which acts on it without global fixed points. An example of such group is Thompson's group F.
In this talk, given a locally moving group G, I will investigate rigidity and flexibility properties of the possible actions of G on the line. It turns out that many locally moving groups (and in particular Thompson's groups F) admit rich (uncountable) families of ``exotic'' actions which are not semi-conjugate to their ``natural'' locally moving action. After giving some examples, I will discuss a result showing all such actions satisfy a specific type of topological dynamical behaviour. Among applications, we will see that if G is locally moving, then all its actions on the real line by C^1-diffeomorphisms must be semi-conjugate to its locally moving action, and that under some additional conditions, a locally moving action is structurally stable under small deformations.
This is a joint work with Joaquín Brum, Cristóbal Rivas and Michele Triestino.