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

Veech surfaces with nonperiodic directions in the trace field


Institut de Mathématiques de Luminy (UPR 9016), 163 Avenue de Luminy, case 907, 13288 Marseille cedex 09, France


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

Chris Johnson, Martin Schmoll. Pseudo-Anosov eigenfoliations on Panov planes. Electronic Research Announcements, 2014, 21: 89-108. doi: 10.3934/era.2014.21.89


Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239


S. Öykü Yurttaş. Dynnikov and train track transition matrices of pseudo-Anosov braids. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 541-570. doi: 10.3934/dcds.2016.36.541


Hieu Trung Do, Thomas A. Schmidt. New infinite families of pseudo-Anosov maps with vanishing Sah-Arnoux-Fathi invariant. Journal of Modern Dynamics, 2016, 10: 541-561. doi: 10.3934/jmd.2016.10.541


Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817


W. Patrick Hooper. An infinite surface with the lattice property Ⅱ: Dynamics of pseudo-Anosovs. Journal of Modern Dynamics, 2019, 14: 243-276. doi: 10.3934/jmd.2019009


Kaushik Nath, Palash Sarkar. Efficient arithmetic in (pseudo-)mersenne prime order fields. Advances in Mathematics of Communications, 2022, 16 (2) : 303-348. doi: 10.3934/amc.2020113


John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369


Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407


Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453


Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005


João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837


Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581


Frédéric Naud, Anke Pohl, Louis Soares. Fractal Weyl bounds and Hecke triangle groups. Electronic Research Announcements, 2019, 26: 24-35. doi: 10.3934/era.2019.26.003


The Editors. William A. Veech's publications. Journal of Modern Dynamics, 2019, 14: i-iv. doi: 10.3934/jmd.2019i


Hiroyuki Kobayashi, Shingo Takeuchi. Applications of generalized trigonometric functions with two parameters. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1509-1521. doi: 10.3934/cpaa.2019072


Meera G. Mainkar, Cynthia E. Will. Examples of Anosov Lie algebras. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 39-52. doi: 10.3934/dcds.2007.18.39


Ferrán Valdez. Veech groups, irrational billiards and stable abelian differentials. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1055-1063. doi: 10.3934/dcds.2012.32.1055


Pascal Hubert, Gabriela Schmithüsen. Infinite translation surfaces with infinitely generated Veech groups. Journal of Modern Dynamics, 2010, 4 (4) : 715-732. doi: 10.3934/jmd.2010.4.715


Giovanni Forni, Howard Masur, John Smillie. Bill Veech's contributions to dynamical systems. Journal of Modern Dynamics, 2019, 14: v-xxv. doi: 10.3934/jmd.2019v

2021 Impact Factor: 0.641


  • PDF downloads (138)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]