Definable topological dynamics and o-minimality

Fix a model $M$. For an $M$-definable group $G$ acting definably and transitively on a definable set $X$, we can consider the induced action on the space $S_X(M)$ of types on $X$. This is an action by homeomorphisms (where $S_X(M)$ is equipped with the standard Stone space topology), making the pair $(G(M),S_X(M))$ a $G(M)$-flow in the sense of classic topological dynamics. I will discuss how various notions of topological dynamics are interpreted in the sense of model theory. I will then present the results on the universal definable flows of groups definable in an o-minimal setting (e.g. definable real Lie groups).