  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.
