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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 Teichmüller theory for Riemann surfaces 11J70 Continued fractions and generalizations
##### Keywords:
Veech surface; pseudo-Anosov; Hecke group; trigonometric fields
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