The rel leaf and real-rel ray of the Arnoux-Yoccoz surface in genus 3

We analyze the rel leaf of the Arnoux-Yoccoz translation surface in genus 3. We show that the leaf is dense in the stratum $\mathcal{H}(2,2{)}^{\mathrm{o}\mathrm{d}\mathrm{d}}$ but that the real-rel trajectory of the surface is divergent. On this real-rel trajectory, the vertical foliation of one surface is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but the vertical foliations on all other surfaces are completely periodic.