  We analyze the rel leaves of the Arnoux-Yoccoz translation surfaces. We show
that for any genus $g \geq 3$, 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 $n \geq 3$, the field extension of the rationals obtained by adjoining
a root of $X^n-X^{n-1}-\ldots-X-1$ has no totally real subfields other than the
rationals.
