A major challenge in dynamics is to classify invariant measures for flows, and especially, to prove unique ergodicity. We will survey two classical arguments in which measure classification has been established. The first argument is an `orbit matching’ argument which was used by Masur in 1982 to establish unique ergodicity of straightline flows on translation surfaces. The second is an argument used by Dani in 1981 to prove unique ergodicity of horocycle flows on compact quotients of SL(2,R). In both arguments, a crucial role is played by a hyperbolic flow which renormalizes the flow being analyzed. We will emphasize the geometric feature that these two arguments in common, and discuss some of the technical issues that need to overcome in order to implement them.
Billiards constitute classic and rich examples of dynamical systems. The trajectories exhibit a wide range of behaviors. The study of rationals billiards leads to the study of translation surfaces and their moduli spaces: this is a bigger space where one can deform the billiard. One very important tool in this deformation theory is the SL(2,R) action on the moduli spaces. We will explain how one can use this action to prove results on billiards. This will be the occasion to revisit Masur's criterion for compact surfaces, and an avatar of Masur's criterion for a non compact billiard: the windtree model.