VEECH SURFACES WITH NONPERIODIC DIRECTIONS IN THE TRACE FIELD

Veech's original examples of translation surfaces V-q with what McMullen has dubbed "optimal dynamics" arise from appropriately gluing sides of two copies of the regular q-gon, with q >= 3. We show that every V-q whose trace field is of degree greater than 2 has nonperiodic directions of vanishing SAF-invariant. (Calta-Smillie have shown that under appropriate normalization, the set of slopes of directions where this invariant vanishes agrees with the trace field.) Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant. In an appendix, we prove various elementary results on the inclusion of trigonometric fields.