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Veech surfaces with nonperiodic directions in the trace field. (English) Zbl 1186.37050
Summary: Veech’s original examples of translation surfaces \(\mathcal V_q\) with what McMullen has dubbed “optimal dynamics” arise from appropriately gluing sides of two copies of the regular \(q\)-gon, with \(q \geq 3\). We show that every \(\mathcal V_q\) 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.

MSC:
37D99 Dynamical systems with hyperbolic behavior
30F60 Teichm├╝ller theory for Riemann surfaces
11J70 Continued fractions and generalizations
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