In these lectures, we will report on some properties of the space of actions of a left-orderable group on the real line. We will notably describe the almost-periodic actions, the harmonic actions and their spaces.

I will report on some joint work with Aurélien Alvarez, which shows that the Jouanolou foliation in degree two is structurally stable, and that it has a non-trivial domain of discontinuity. This result is opposed to a series of results beginning in the sixties with the works of Hudai-Verenov and Ilyashenko.