Isoperiodic dynamics in rank 1 affine invariant orbifolds

TL;DR

This paper investigates the structure of isoperiodic foliations in rank 1 affine invariant orbifolds, proving they are either all closed or all dense, and establishing ergodicity in the dense case.

Contribution

It provides a dichotomy for the leaves of the isoperiodic foliation and proves ergodicity for dense leaves in rank 1 affine invariant orbifolds.

Findings

Leaves are either all closed or all dense.

Ergodicity holds for dense leaves with respect to the affine measure.

The dichotomy clarifies the global structure of isoperiodic foliations in this setting.

Abstract

Let M be a rank 1 affine invariant orbifold in a stratum of the moduli space of flat surfaces. We show that the leaves of the M-isoperiodic foliation are either all closed or all dense. In the second case, we establish ergodicity of the foliation with respect to the affine measure on M.