Rel leaves of the Arnoux–Yoccoz surfaces

We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus g⩾ 3 , the leaf is dense in the connected component of the stratum H(g- 1 , g- 1) to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux–Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any n⩾ 3 , the field extension of Q obtained by adjoining a root of Xⁿ- X^n-1- ⋯ - X- 1 has no totally real subfields other than Q.