Rel leaves of the Arnoux–Yoccoz surfaces

• W. Patrick Hooper ^[1] ; Barak Weiss ^[2]
  1. [1] City College of New York

     City College of New York

     Estados Unidos

     
  2. [2] Tel Aviv University

     Tel Aviv University

     Israel

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 2, 2018, págs. 875-934
• Idioma: inglés
• Enlaces
  • Texto completo (pdf)
• Resumen

  • We analyze the rel leaves of the Arnoux–Yoccoz translation surfaces. We show that for any genus g⩾3 , the leaf is dense in the connected component of the stratum H(g−1,g−1) to which it belongs, and the one-sided imaginary-rel trajectory of the surface is divergent. For one surface on this trajectory, namely the Arnoux–Yoccoz surface itself, the horizontal foliation is invariant under a pseudo-Anosov map (and in particular is uniquely ergodic), but for all other surfaces, the horizontal foliation is completely periodic. The appendix proves a field theoretic result needed for denseness of the leaf: for any n⩾3 , the field extension of Q obtained by adjoining a root of Xn−Xn−1−⋯−X−1 has no totally real subfields other than Q .