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Veech surfaces with nonperiodic directions in the trace field

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  • 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 $q \ge 3$. 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.
    Mathematics Subject Classification: Primary: 37D99, 30F60; Secondary: 11J70.

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