A non-zero Abelian differential in a compact Riemann surface of genus $g \geq 1$ endows the surface with an atlas (outside the zeroes) whose coordinate changes are translations. There is a natural ``vertical flow'' (moving up with unit speed) associated with the translation structure, generalizing the genus $1$ case of irrational flows on the torus.

The Teichm\"uller flow in the moduli space of Abelian differentials can be seen as the renormalization operator of translation flows. In this talk, we will discuss how the chaoticity of the Teichm\"uller flow dynamics reflects on the (non-chaotic) dynamics of the associated vertical flows (for typical parameters), and the closely related interval exchange transformations.