# American Institute of Mathematical Sciences

October  2009, 3(4): 611-629. doi: 10.3934/jmd.2009.3.611

## Veech surfaces with nonperiodic directions in the trace field

 1 Institut de Mathématiques de Luminy (UPR 9016), 163 Avenue de Luminy, case 907, 13288 Marseille cedex 09, France 2 Oregon State University, Corvallis,OR 97331, United States

Received  September 2009 Published  January 2010

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.
Citation: Pierre Arnoux, Thomas A. Schmidt. Veech surfaces with nonperiodic directions in the trace field. Journal of Modern Dynamics, 2009, 3 (4) : 611-629. doi: 10.3934/jmd.2009.3.611
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