Suppose you have a group of transformations of a space. If you know something about individual transformations, can you extrapolate to say something global about the whole system? An old example of this is a theorem of Hölder: if you have a group of homeomorphisms of the real line and none of them fixes a point, then the group is abelian and the whole system conjugate to an action by translations.
In this talk, I'll introduce you to this family of problems and explain recent joint work with Thomas Barthelmé that proves a new such result about groups acting on the line. As an application, we use this to prove rigidity results for a different, fascinating family of dynamical systems, Anosov flows in dimension 3.