A criterion for density of the isoperiodic leaves in rank one affine invariant orbifolds

We define on any affine invariant orbifold M$\mathcal {M}$ a foliation FM$\mathcal {F}^{\mathcal {M}}$ that generalizes the isoperiodic foliation on strata of the moduli space of translation surfaces and study the dynamics of its leaves in the rank 1 case. We establish a criterion that ensures the density of the leaves and provide two applications of this criterion. The first one is a classification of the dynamical behavior of the leaves of FM$\mathcal {F}^{\mathcal {M}}$ when M$\mathcal {M}$ is a connected component of a Prym eigenform locus in genus 2 or 3 and the second provides the first examples of dense isoperiodic leaves in the stratum H(2,1,1)$\mathcal {H}(2,1,1)$ .