Infinitely many lattice surfaces with special pseudo-Anosov maps
Kariane Calta Thomas A. Schmidt
Journal of Modern Dynamics 2013, 7(2): 239-254 doi: 10.3934/jmd.2013.7.239
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of nonparabolic elements in the periodic field of certain translation surfaces.
keywords: translation surface flux Veech group. Pseudo-Anosov SAF invariant
New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant
Hieu Trung Do Thomas A. Schmidt
Journal of Modern Dynamics 2016, 10(02): 541-561 doi: 10.3934/jmd.2016.10.541
We show that an orientable pseudo-Anosov homeomorphism has vanishing Sah-Arnoux-Fathi invariant if and only if the minimal polynomial of its dilatation is not reciprocal. We relate this to works of Margalit-Spallone and Birman, Brinkmann and Kawamuro. Mainly, we use Veech's construction of pseudo-Anosov maps to give explicit pseudo-Anosov maps of vanishing Sah-Arnoux-Fathi invariant. In particular, we give a new infinite family of such maps in genus 3.
keywords: Pisot units Pseudo-Anosov translation surfaces. bi-Perron units Sah-Arnoux-Fathi invariant
Veech surfaces with nonperiodic directions in the trace field
Pierre Arnoux Thomas A. Schmidt
Journal of Modern Dynamics 2009, 3(4): 611-629 doi: 10.3934/jmd.2009.3.611
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.
Commensurable continued fractions
Pierre Arnoux Thomas A. Schmidt
Discrete & Continuous Dynamical Systems - A 2014, 34(11): 4389-4418 doi: 10.3934/dcds.2014.34.4389
We compare two families of continued fractions algorithms, the symmetrized Rosen algorithm and the Veech algorithm. Each of these algorithms expands real numbers in terms of certain algebraic integers. We give explicit models of the natural extension of the maps associated with these algorithms; prove that these natural extensions are in fact conjugate to the first return map of the geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost every real number has an infinite number of common approximants for both algorithms.
keywords: Moebius transformations Fuchsian groups. geodesic flow Continued fractions natural extension

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