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

Veech surfaces with nonperiodic directions in the trace field

Pierre Arnoux 1, and  Thomas A. Schmidt 2,

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.
Keywords: Veech surface, Hecke group, pseudo-Anosov, trigonometric fields..
Mathematics Subject Classification: Primary: 37D99, 30F60; Secondary: 11J7.
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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