Isoperiodic dynamics in rank 1 affine invariant orbifolds

The paper proves: For any rank‑1 affine invariant orbifold M, the M‑isoperiodic leaves are either all closed or all dense; in the dense case the foliation is ergodic for the affine SL2(R)‑invariant probability measure. This is stated in the abstract and Theorem A and proved by constructing a measure‑theoretic leaf space P(M1), producing an SL2(R)‑equivariant map π′, showing essential transitivity of the induced action on P(M1), and analyzing the closed finite‑covolume stabilizer H to obtain the closed/dense dichotomy via Borel density; ergodicity follows when H = SL2(R) (paper: Proposition 1, Propositions 2–4, and the proof of Theorem A) . The candidate solution reproduces this strategy: conditional measures along leaves, an (almost) equivariant map to P(M1), essential transitivity and a stabilizer H of finite covolume, and the same case split. The only notable slip is identifying the leaf space with SL2(R)/H (where H is the stabilizer of the probability measure), whereas the paper correctly uses SL2(R)/Stab(leaf) to conclude Hausdorffness and closed leaves; however, since Stab(leaf) ≤ H and H is a lattice, Stab(leaf) is discrete, and the intended conclusion still follows. Aside from this point and some minor omissions in measurability/equivariance justification (addressed by π′ in the paper), the reasoning matches the paper’s proof structure and conclusions.