Veech surfaces with nonperiodic directions in the trace field

Pierre Arnoux; Thomas A. Schmidt

doi:10.3934/jmd.2009.3.611

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2009, Volume 3, Issue 4: 611-629. Doi: 10.3934/jmd.2009.3.611

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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
2.
Oregon State University, Corvallis,OR 97331

Received: September 2009

Published: January 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 37D99, 30F60; Secondary: 11J70.

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