Dynamics of the absolute period foliation of a stratum of holomorphic 1-forms

Let $\mathcal{C}$ be a connected component of a stratum of the moduli space of holomorphic $1$-forms of genus $g$. We show that the absolute period foliation of $\mathcal{C}$ is ergodic on the area-$1$ locus, and that the non-dense leaves lie in an explicit countable union of suborbifolds, subject to a mild assumption on $\mathcal{C}$. We show similar results for subspaces of $\mathcal{C}$ defined by topological restrictions on the absolute periods. We obtain these dynamical results by showing that for a typical positive cohomology class in $H^1(S_g;\mathbb{C})$, the associated space of isoperiodic forms in $\mathcal{C}$ is connected. Lastly, we show that certain covering constructions provide examples of spaces of isoperiodic forms with positive dimension and infinitely many connected components.