In this talk, we will typically consider a homogeneous polynomial vector field $X$ of degree $\geq 2$ on ${\mathbb C}^2$. In particular, the time-one map induced by $X$ defines a germ of parabolic diffeomorphism $h$ of $(C^2,0)$. The vector field $X$ also induces a singular affine structure on the Riemann sphere which, in turn, leads to a "geodesic flow", or "billiard dynamics", encoding much of the dynamics of $h$. We will try to make these connections accurate, in particular explaining how the monodromy of the mentioned affine structure can be read off the projective holonomy of the foliation associated with $X$. If time permits, we might say a word about higher dimensional versions of this construction.
